Heights and multiplicative relations on algebraic varieties · varieties are defined over Q, the field of algebraic numbers. One can define a height function on the set of algebraic

Heights and multiplicativerelations on algebraic varieties

Inauguraldissertationzur

Erlangung der Wurde eines Doktors der Philosophie

vorgelegt derPhilosophisch-Naturwissenschaftlichen Fakultat

der Universitat Basel

von

Philipp Habegger

ausTrub BE

Basel, 2007

Genehmigt von der Philosophisch-Naturwissenschaftlichen Fakultat aufAntrag von Prof. Dr. D.W. Masser und Prof. Dr. S. David.

Basel, den 26. Juni 2007

Prof. Dr. Hans-Peter HauriDekan

Contents

Acknowledgements 3

Introduction 5

Chapter 1. A review of heights 111. The Weil height 112. Height and Mahler measure of a polynomial 13

Chapter 2. Multiplicative dependence and isolation I 171. Height bounds for dependent solutions of x + y = 1 172. Proof of Theorem 2.1 183. Proof of Theorem 2.2 20

Chapter 3. Multiplicative dependence and isolation II 231. Introduction 232. Factorizing trinomials over Kroneckerian fields 263. A reduction 284. Averaging the Mahler measure 295. Bounding the variation 316. Bounding the discrepancy 367. Proof of Theorem 3.1 and Corollary 3.3 378. Counting multiplicative dependent points 40

Chapter 4. Dependent solutions of x + y = α 431. Height bounds for dependent solutions of x + y = α 432. Notation and auxiliary results on rational functions 463. Proofs of Propositions 4.1, 4.2, 4.3 and Corollary 474. Proofs of Theorems 4.1 and 4.2 545. Dependent solutions with large height 556. Proof of Theorem 4.4 567. Proof of Theorem 4.5 598. Proof of Theorem 4.6 62

Chapter 5. More on heights and some algebraic geometry 651. Some preliminaries 652. Chow forms and higher dimensional heights 66

1

2 CONTENTS

3. Normalized height and essential minimum of a variety 68

Chapter 6. Intersecting varieties with small subgroups 711. Introduction 712. Algebraic subgroups of Gn

m 763. Some geometry of numbers 774. Proof of Theorem 6.1 79

Chapter 7. A Bogomolov property modulo algebraic subgroups 851. Introduction 852. Auxiliary results 893. Push-forwards and pull-backs 914. A lower bound for the product of heights 925. Proof of Theorem 7.1 and corollaries 96

Appendix A. Quasi-equivalence of heights 1031. Introduction 1032. Construction via Siegel’s Lemma 1053. Zero bounds 1094. Completion of proof 113

Appendix B. Applications of the Quasi-equivalence Theorem 1191. On a Theorem of Bombieri, Masser, and Zannier 1192. On a Theorem of Runge 1213. On a Theorem of Skolem 1234. On a Theorem of Sprindzhuk 127

Bibliography 129

Curriculum Vitae 133

Acknowledgements

I would like to thank my adviser Prof. David W. Masser for introducing me intothe active field of research revolving around the theory of heights. His large experience,his many insightful comments, and his support have helped me greatly throughout myyears as a PhD student. He also carefully read this manuscript and gave many invaluablesuggestions for improvement.

Many thanks go also to Prof. Sinnou David who agreed to referee my thesis, whohelped me organize my six month visit the Insitut de Mathematiques de Jussieu in Paris,and with whom I had many interesting discussions there. I enjoyed my stay in Parisvery much, both from a mathematical and cultural point of view.

Furthermore, I would like to thank Prof. Francesco Amoroso, Prof. Michael Naka-maye, Prof. Patrice Philippon, Prof. Gael Remond, Prof. Evelina Viada, and Prof.Umberto Zannier for the fruitful conversations I had over the years. I am gratefulfor Prof. Zannier’s comments on an early version of the chapter containing the Quasi-equivalence result.

During my thesis I received financial support from a Schweizer Nationalfonds grant.I obtained financial support from the Department of Mathematics in Basel directlybefore and after my SNF grant. While in Paris I received financial support from theInstitut de Mathemathiques de Jussieu.

In the last few years I had the pleasure to attend several conferences among them theDiophantine Geometry Research Period at the Scuola Normale Superiore di Pisa, theDiophantine Approximation and Heights Program at the ESI in Vienna, Approximationdiophantienne et nombres transcendants at the CIRM in Luminy, and the WorkshopDiophantische Approximationen in Oberwolfach. I would like to thank the organizers ofthese conferences for inviting me and for giving me the possibility to present my work.

Thanks go also to my friends, Em, Giuliana, Jonas, Martin, Patrick, Primo, Utefrom inside and outside the University for keeping my spirits high and of course for theendless discussions during coffee break. I would like to especially thank Brita for thewonderful time we spent together in Basel, Berlin, and Paris.

Finally, I thank my parents, they were always there for me in the inevitable ups anddowns in life. I owe this thesis to my mother who always supported my decision in lifeand my father who sparked my interest in mathematics.

3

Introduction

Points on a subvariety X of a semi-abelian variety A that are contained in a sub-group, let the subgroup be of finite rank or algebraic, are subject to severe restrictionsarithmetical nature.

Finiteness results for intersections of X with subgroups of finite rank have been stud-ied by Faltings, Hindry, Laurent, McQuillan, Raynaud, Vojta and others. More recentlyseveral authors ([CZ00], [BMZ99], [BMZ03], [BMZ06a], [BMZ06b], [BMZ04],[Via03], [RV03], [Rem05b], [Rem07], [Pin05b], [Zan00], [Zil02], [Mau06]) haveconsidered the intersection of X with A[r], the set of complex points in A contained inan algebraic subgroup of codimension greater or equal to r. If H is a fixed algebraicsubgroup of A with codimension strictly less than dim X, then a dimension countingargument shows that X∩H is either empty or contains a curve. As we are allowing H tovary with fixed codimension, the intersection X ∩A[r] may be quite large if r < dim X.In this thesis we are only interested in the case r ≥ dim X.

If not stated otherwise we will also assume throughout the introduction that allvarieties are defined over Q, the field of algebraic numbers. One can define a heightfunction on the set of algebraic points of A. Throughout this thesis we work only inthe algebraic torus Gn

m or an abelian variety. So we can take the Weil height or theNeron-Tate height associated to an ample line bundle.

We will pursue two types of questions. First, for which r does the set X ′(Q) ∩ A[r]

have bounded height and how do these bounds depend on X? Second, for which r is theset X ′′(Q) ∩ A[r] finite? Here X ′ and X ′′ are obtained from removing from X certainsubvarieties in order to to eliminate trivial counterexamples. For example if X is aproper algebraic subgroup of Gn

m with positive dimension, then there is no hope for aboundedness of height or finiteness result for U(Q) ∩ (Gn

m)[r] if r ≤ dim X and if U isZariski open and dense in X. In this case X ′ and X ′′ are both empty.

The simplest non-trivial example seems to be the curve defined by x + y = 1 inG2

m. Here we can take X ′ and X ′′ to equal our curve. Algebraic subgroups of G2m

can be described by at most two monomial relations xαyβ = 1 with integer exponentsα and β. For subgroups of dimension 1, one non-trivial relation suffices. If (x, y) iscontained in such a subgroup then x and y are called multiplicatively dependent. Hencethe intersection of our curve with the union of all proper algebraic subgroups of G2

m canbe described by the solutions of

(0.0.1) xα(1− x)β = 1.

5

6 HEIGHTS AND MULTIPLICATIVE RELATIONS ON ALGEBRAIC VARIETIES

This is an equation in three unknowns x, α, and β, so one should not expect finitelymany solutions. Indeed, taking x 6= 1 a root of unity gives infinitely many solutions.

In [CZ00] Cohen and Zannier showed that if H denotes the absolute non-logarithmicWeil height then (0.0.1) implies the sharp inequality max{H(x),H(1−x)} ≤ 2. In chap-ter 2 we start off by giving an alternative proof of Cohen and Zannier’s Theorem. Weeven show that the possibly larger height H(x, 1−x) is at most 2. In their paper, Cohenand Zannier also proved that 2 is an isolated point in the range of max{H(x),H(1−x)}.We make this result explicit in Theorem 2.2, working instead with H(x, 1 − x). Theproof applies Smyth’s Theorem on lower bounds for heights of non-reciprocal algebraicnumbers and a Theorem of Mignotte.

As was already noticed in [CZ00], solutions of (0.0.1) are closely linked to roots ofcertain trinomials whose coefficients are roots of unity. In chapter 3 Theorem 3.2 wefollow this avenue by factoring such trinomials over cyclotomic fields. Having essentiallya minimal polynomial in our hands, we obtain a new proof for the boundedness ofH(x, 1 − x) with x as in (0.0.1). More importantly, in Theorem 3.1 we show that notonly is 2 isolated in the range of the height function, but also that H(x, 1−x) convergesto an absolute constant if [Q(x) : Q] goes to infinity. The proof determines the valueof this limit: it is the Mahler measure of the two-variable polynomial X + Y − 1. In acertain sense this Mahler measure is the height of the curve in our problem.

In Theorem 3.3 we prove a conjecture of Masser stated in [Mas07]: the numberof solutions of (0.0.1) with [Q(x) : Q] ≤ D is asymptotically equal to c0D

3 with c0 =2.06126 . . . as D →∞. The constant c0 is defined properly in chapter 3 as a convergingseries. This counting result is a further application of Theorem 3.2.

In chapter 4 we generalize the method from chapter 2 to bound the height of mul-tiplicatively dependent solutions of

(0.0.2) x + y = α.

Here α is now any non-zero algebraic number. In [BMZ99] Bombieri, Masser, andZannier prove a more general result which also implies boundedness of height in thiscase. Their Proposition A leads to an explicit upper bound for the height; the boundis polynomial in H(α). We are mainly interested in upper bounds for H(x, y) whichhave good dependency in H(α). The value H(α) can be regarded as the height ofthe defining equation (0.0.2). In Theorems 4.1 and 4.2 we get the bound H(x, y) ≤2H(α) min{H(α), 7 log(3H(α))}. By Theorem 4.3 the exponent of the logarithm cannotbe less than 1. But in some special cases, e.g. if α is a rational integer, we improve theupper bound to 2H(α), see Theorem 4.4. In this theorem we also show that if α is arational integer then 2H(α) is attained as a height if and only if α is a power of two.Thus if α is a power of two, then our bound is sharp. For such α and if also α ≥ 2 weprove in Theorem 4.5 that 2H(α) is isolated in the range of the height.

Starting from chapter 6 we work in an algebraic torus of arbitrary dimension. Alge-braic subgroups can still be described by a finite set of monomial equations. For example(x1, . . . , xn) ∈ Gn

m(C) is contained in a proper algebraic subgroup if and only if the xi

INTRODUCTION 7

satisfy a non-trivial multiplicative relation. In [BMZ99] Bombieri, Masser, and Zan-nier proved that if X is an irreducible curve which is not contained in the translate of aproper algebraic subgroup, then points on X that lie in a proper algebraic subgroup havebounded height. Moreover, they showed that this statement is false if X is contained inthe translate of a proper algebraic subgroup. The authors also showed that there areonly finitely many points on X that lie in an algebraic subgroup of codimension at least2. This finiteness result was generalized by the same authors in [BMZ03] to algebraiccurves defined over the field of complex numbers. Hence for curves it makes sense totake X ′ = X if X is not contained in the translate of a proper algebraic subgroup andX ′ = ∅ else wise. But X ′′ is more subtle: we take X ′′ = X if X is not contained in aproper algebraic subgroup and X ′′ = ∅ else wise. The point in making this distinction isthat in [BMZ06a] the authors conjectured that X ′′ contains only finitely many pointsin an algebraic subgroup of codimension at least 2. They proved this conjecture forn ≤ 5. Recently, in [Mau06] Maurin gave a proof for all n.

Let X ⊂ Gnm be an irreducible subvariety, not necessarily a curve. In the higher

dimensional case we finally need a definition of X ′: we get X ′ by removing from Xall positive dimension subvarieties that show up in an improper component of the in-tersection of X with the translate of an algebraic subgroup. The definition of X ′′ issimilar but we require the translates of algebraic subgroups to be algebraic subgroups.In [BMZ06b] Bombieri, Masser, and Zannier showed that X ′ is Zariski open in X.

Let h be the absolute logarithmic Weil height. Our contribution in chapter 6 isTheorem 6.1 where we give an explicit bound for the height of algebraic points p in X ′

that lie “uniformly close” to an algebraic subgroup of codimension strictly greater thann− n/ dim X. By uniformly close we mean that there exist an ε > 0, independent of p,and an a in an algebraic subgroup of said codimension with h(pa−1) ≤ ε. Actually, inTheorem 6.1 we will use a weaker notion of uniformly close. The terminology comes fromthe fact that the map (p, a) 7→ h(pa−1) has similar properties as a distance function.For example it satisfies the triangle inequality. This notion of distance was consideredby several authors ([Eve02], [Poo99], [Rem03]) in connection with subgroups of finiterank.

Theorem 6.1 generalizes the Bounded Height Theorem for curves by Bombieri,Masser, and Zannier. We state our theorem such that it also gives an explicit versionof a Theorem of Bombieri and Zannier in [Zan00] on the intersection of varieties withone dimensional subgroups. To do this we will need a slightly more general definitionof X ′ which is provided in chapter 6.

The height upper bound in Theorem 6.1 involves, along with n, the degree andheight of the variety X. We define these two notions in chapter 5. In simple terms, theheight of X controls the heights of the coefficients of a certain set of defining equationsfor X whereas the degree of X controls their degrees. Just as in the second proof forheight bounds on curves given in [BMZ99], our proof of Theorem 6.1 uses ideas fromthe geometry of numbers. Given p ∈ X(Q) uniformly close to an algebraic subgroup weconstruct a new algebraic subgroup H of codimension dim X and controlled degree, suchthat pH has normalized height small compared to the height of p. We then intersect


pH with X. The Arithmetic Bezout Theorem bounds the height of isolated points inthis intersection leading to an explicit height bound for p.

Lehmer-type lower bounds for heights in spirit of Dobrowolski’s Theorem and itsgeneralization to higher dimension provide a method for deducing finiteness results fromheight bounds as given in chapter 6. This method was used together with algebraic num-ber theory in Bombieri, Masser, and Zannier’s article [BMZ99] to prove the finitenessof the set of points on X ′ in an algebraic subgroup of codimension at least 2 if X is acurve. Meanwhile, their intricate argument has been simplified in [BMZ04] by apply-ing a more advanced height lower bound due to Amoroso and David [AD04]. In thislower bound the degree over Q of a point is essentially replaced by its degree over themaximal abelian extension of Q. Using this approach we show in Corollary 6.2 that ifX is a surface in G5

m, then there are only finitely many points on X ′ contained in analgebraic subgroup of codimension at least 3. Thus we have finiteness for the correctsubgroup size at least in an isolated case.

Even in presence of a uniform height bound as in Theorem 6.1, the approaches in[BMZ99] and [BMZ04] cannot be used to prove the finiteness of the set of p ∈ X ′(Q)with h(pa−1) small and a contained in an algebraic subgroups of appropriate dimension:although pa−1 has small height, its degree cannot be controlled. In chapter 7 we pursuea new approach using a Bogomolov-type height lower bound. This bound was proved byAmoroso and David in [AD03]; it bounds from below the height of a generic point on avariety not equal to the translate of an algebraic subgroup. The main result of chapter7 is Theorem 7.1: we show that for B ∈ R there exists an ε = ε(X, B) > 0 with thefollowing property: there are only finitely many p ∈ X ′(Q) with h(pa−1) ≤ ε where a iscontained in an algebraic subgroup of dimension strictly less than m(dim X, n). In otherwords, there are only finitely many algebraic points on X ′ of bounded height which areuniformly close to an algebraic subgroup of dimension less than m(dim X, n). Just aswas the case in Theorem 6.1 we actually use a relaxed version of uniformly close inTheorem 7.1. The somewhat unnatural function m(·, ·) is defined in (7.1.1). At least inthe case of curves we have n − 2 < m(1, n) and so we can take the subgroups to havethe best possible dimension n− 2. Unfortunately this is the only interesting case wherem(r, n) > n− r − 1.

With the height upper bound from chapter 6 we can deduce a corollary to Theorem7.1 which proves finiteness independently of B and where the subgroup dimension isstrictly less than min{n/ dim X, m(dim X, n)}. Let X be a curve, then this result isoptimal with respect to the subgroup dimension. Let us assume that X is not containedin the translate of a proper algebraic subgroup, hence X ′ = X. Then our corollary saysthat there are only finitely many algebraic points on X that are close to an algebraicsubgroup of codimension at least 2. Moreover, in Corollary 7.2 we use Dobrowolski’sTheorem to show that if ε in the definition of uniformly close is small enough, thenall points on X close to an algebraic subgroup of codimension at least 2 are actuallycontained in such a subgroup.

INTRODUCTION 9

We now shift our focus from the algebraic torus to abelian varieties: we want to studythe intersection X ′(Q)∩A[r] where A is an abelian variety and X is an irreducible closedsubvariety of A. The definitions of X ′ and X ′′ make sense in the abelian setting andare completely analog to the multiplicative case.

Let X be a curve, then in [Via03] Viada proved that X ′(Q)∩A[1] has bounded heightif A is a power of an elliptic curve. If the elliptic curve has complex multiplication shealso proved that X ′(Q)∩A[2] is finite. Remond in [Rem05b] generalized Viada’s heightbound to any abelian variety. In [Rem07] Remond applied a generalization of Vojta’sinequality which he proved in [Rem05a] and in Theorem 1.2 showed boundedness ofheight of (X(Q)\Z(r)

X )∩A[r]. Here X\Z(r)X ⊂ X is a new deprived subset which depends

on r. In fact his result holds for a set larger than A[r] involving also the division closureof finitely generated group. If A is isogenous to a product of elliptic curves and if X is asufficiently general surface which is not contained in the translate of a proper algebraicsubgroup then X\Z(r)

X is non-empty and Zariski open in X for r ≥ (dim A + 3)/2.In [RV03], Remond and Viada proved that if X is a curve then X ′′(Q) ∩ A[2] is

finite if A is a power of an elliptic curve E with complex multiplication. In a recentpreprint, Viada [Via07] announced the finiteness of X ′′(Q) ∩ A[3] for unrestricted E,the optimal subgroup codimension 2 is thus just missed.

We announce the following result called the Bounded Height Theorem: if A = Eg

is a power of an elliptic curve E and X is an irreducible closed subvariety of arbitrarydimension, then X ′(Q) ∩ A[dim X] has bounded Neron-Tate height. Also, using a resultfrom Kirby’s Thesis [Kir06] and ideas from Bombieri, Masser, and Zannier’s [BMZ06b]one can show that X ′ is Zariski open and give a criterion on X to decide when X ′ isnon-empty. Using height lower bounds on abelian varieties with complex multiplicationdue to Ratazzi in [Rat07] we can use the Bounded Height Theorem to show thatX ′(Q)∩A[dim X+1] is finite if E has complex multiplication. For an elliptic curve withoutcomplex multiplication, finiteness of X ′(Q)∩A[r] can also be obtained, using for exampleRemond’s Theorem 2.1 from [Rem05b]. But r is in general sub-optimal for such ellipticcurves.

The essential difference between the Bounded Height Theorem and Theorem 6.1 isthat the subgroups are now allowed to have the best-possible codimension dim X for allX.

In the future we plan to publish these results.Pink has stated a general conjecture on mixed Shimura varieties, see [Pin05a] and

[Pin05b]. One special implication is his Conjecture 5.1 from [Pin05b]: if A is a semi-abelian variety defined over C and if X ⊂ A is a subvariety also defined over C which isnot contained in a proper algebraic subgroup of A, then X(C)∩A[dim X+1] is not Zariskidense in X. Zilber’s stronger Conjecture 2 in [Zil02] implies the same conclusion. Withthe Bounded Height Theorem we can prove this assertion under the following strongerhypothesis on A and X: A is a power of an elliptic curve E with complex multiplicationand if ϕ : Eg → Edim X is a surjective homomorphism of algebraic groups, then therestriction ϕ|X : X → Edim X is dominant.


The proof of the Bounded Height Theorem uses the completeness of A (and X)in an essential way as it relies on intersection theory. Nevertheless, a proof for theboundedness of height of X ′(Q)∩A[dim X] for the non-complete X ⊂ A = Gn

m along thelines of the proof of the Bounded Height Theorem must not be ruled out. For instanceone could compactify Gn

m ↪→ Pn and work in Pn. Still, there seems to be no suitableTheorem of the Cube for Gn

m. Future research could consist in finding a proof of theBounded Height Theorem in the multiplicative case or in abelian varieties other than apower of an elliptic curve.

In the two appendices we leave the main path of the thesis. Let P be an irreduciblepolynomial in two variables with algebraic coefficients. Say x and y are algebraic withP (x, y) = 0. In appendix A, motivated by Proposition B of [BMZ99], we consider theproblem of bounding |degX(P )h(x)−degY (P )h(y)| explicitly and with good dependencyin h(x), h(y), and P .

For simple examples such as P = Xp − Y q with p and q coprime integers, theabsolute value is zero. But for general and fixed P it may even be unbounded as (x, y)runs over all algebraic solutions of P . In Theorem A.1 we prove an upper bound which isof the form cmax{1, hp(P )}1/2 max{1, h(x), h(y)}1/2 where the constant c is completelyexplicit and depends only on the partial degrees of P . Here hp(P ) is the projectivelogarithmic Weil height of the coefficient vector of P . This type of height inequality isoften referred to as quasi-equivalence of heights.

In appendix B we demonstrate four known results using the Quasi-equivalence The-orem from appendix A. The first application is the Theorem of Bombieri, Masser, andZannier, already discussed above, in the case of curves in G2

m. We then prove a versionof Runge’s Theorem on the finiteness of the number of solutions of certain diophantineequations. Next we show a result of Skolem from 1929: we first generalize the greatestcommon divisor of pairs of integers to pairs of algebraic numbers. We then show thatif x and y are coprime algebraic numbers and P (x, y) = 0 where P is an irreduciblepolynomial in Q[X, Y ] without constant term, then x and y have uniformly boundedheight. This result has been proved independently by Abouzaid in [Abo06] who used itto prove a variant of the Quasi-equivalence Theorem. The fourth and final applicationis an explicit version of Sprindzhuk’s Theorem: let P have rational coefficients, againwithout constant term and such that not both partial derivatives of P vanish at (0, 0).Then for a sufficiently large prime l, the polynomial P (l, Y ) ∈ Q[Y ] is irreducible. Sincethe Quasi-equivalence Theorem gives explicit bounds, so do its four applications.

Chapters 1 and 5 contain no new results but serve as reference for certain theoremswhich we apply in the rest of the thesis. Chapter 1 introduces the Weil height andrelated subjects. It is used throughout the thesis. Chapter 5 contains some results fromalgebraic geometry and gives a definition for the height of a positive dimensional variety.These definitions and results will be used in the second part of the thesis, chapters 6and 7.

CHAPTER 1

A review of heights

Heights play a central role in this thesis. On the one hand they are an importanttechnical tool to control the “size” of algebraic numbers, often need to prove finitenessresults. On the other hand heights have subtle properties which makes them intrinsicallyinteresting. We dedicate this first chapter to a short review of the absolute Weil height.This particular notation of height will be used often, especially in the first part of thethesis. We present basic functional properties and also some notation which will be freelyused in this work. We will then define the height and Mahler measure of polynomials andalso present some results. Our main reference is Bombieri and Gubler’s book [BG06].This chapter should be seen mainly as a source of reference for later chapters. Thereis not the faintest claim that this chapter gives an complete overview of the topic ofheights. In chapter 5 we will revisit heights, but from a different point of view.

1. The Weil height

Recall that an absolute value |·| on a field K satisfies the triangle inequality |x+y| ≤|x| + |y| for all x, y ∈ K. The absolute value is called ultrametric if it satisfies theultrametric inequality |x + y| ≤ max{|x|, |y|} for all x, y ∈ K.

Let K be a number field with ring of algebraic integers OK . If I ⊂ K is a fractionalideal, we set N(I) ∈ Q to be its norm. We define MK to be the set of absolute valuesof K such that their restriction to Q is the usual p-adic absolute value or the standardcomplex absolute value. Elements of MK will be called places of K. Let v ∈ MK extendw ∈ MQ; we will write v | w. If w is the standard complex absolute value on Q then vwill be called an infinite place, or v | ∞ for short. If w is a p-adic absolute value thenv will be called a finite place, or v - ∞ for short. It is well-known that there are one-to-one correspondences between infinite places and embeddings K → C up to complexconjugation on the one hand and between finite places and non-zero prime ideals of OK

on the other hand. We define the local degree dv = [Kv : Qw] where Kv, Qw are thecompletions of K, Q with respect to the absolute values. For integers n it is sometimesuseful to define δv(n) = max{1, |n|v}.

If τ ∈ K∗ = K\{0}, then there are at most finitely many v ∈ MK such that |τ |v 6= 1.We have the product formula (Proposition 1.4.4 [BG06])

(1.1.1)∏

v∈MK

|τ |dvv = 1 for any τ ∈ K∗.

11


Let p = (p1, . . . , pn) ∈ Kn, the absolute Weil height of p is defined as

(1.1.2) H(p) =∏

v∈MK

max{1, |p1|v, . . . , |pn|v}dv/[K:Q].

This definition is independent of the field K containing the pi ([BG06] Lemma 1.5.2page 15) and the height function is thus defined on Qn, for Q the algebraic closure ofQ. For the reader’s convenience we review some basic properties of the height function:

Say τ ∈ Q has minimal polynomial adTd + · · · + a0 ∈ Q[T ] over Q such that the

ai are integers having no common factor and ad 6= 0. We can calculate the height of τmore directly with the formula

(1.1.3) H(τ)d = |ad|d∏

i=1

max{1, |τi|}

where τ1, . . . , τd ∈ C are the distinct zeros of P . The equality (1.1.3) follows fromPropositions 1.6.5 and 1.6.6 in [BG06].

Say p ∈ Qn and k ∈ N, then

(1.1.4) H(pk) = H(p)k.

The proof of (1.1.4) follows directly from the definition (1.1.2). If τ ∈ Q∗ we can saymore, in fact for k ∈ Z we have

(1.1.5) H(τk) = H(τ)|k|.

This equality follows from (1.1.1) and (1.1.4) for negative k. If p ∈ (Q∗)n, then ingeneral only H(p−1) ≤ H(p)n holds.

Say τ ∈ Q, then

H(τ) = 1 if and only if τ is a root of unity or zero.

This is Kronecker’s Theorem, for a proof see Theorem 1.5.9 in [BG06].The height is invariant under multiplication by a root of unity: if p ∈ Qn and ζ a

root of unity, then H(ζp) = H(p).For τ, µ ∈ Q we may bound

H(τµ) ≤ H(τ)H(µ).

This inequality follows from max{1, |τµ|} ≤ max{1, |τ |}max{1, |µ|} for any absolute | · |on a field containing τ and µ. An analogue bound for sums is

(1.1.6) H(τ + µ) ≤ 2H(τ)H(µ),

for τ, µ ∈ Q. Indeed (1.1.6) follows from

max{1, |τ + µ|} ≤ δ max{1, |τ |}max{1, |µ|}with δ = 1 if | · | satisfies the ultrametric inequality and δ = 2 else wise. If µ = 1 wehave the special case H(1 + τ) ≤ 2H(τ). In general the factor 2 cannot be omitted inthis inequality. But in chapter 2 Lemma 2.4 we will improve 2 to 1.909 . . . for non-zeroτ of small height that are not conjugate to τ−1.

1. A REVIEW OF HEIGHTS 13

If a ∈ Z and b ∈ N are coprime, then

H(a/b) = max{|a|, b}.

This equality follows from (1.1.3).Finally Northcott’s Theorem says that for C,D ∈ R, there are at most finitely many

τ ∈ Q with H(τ) ≤ C and [Q(τ) : Q] ≤ D. For a proof see Theorem 1.6.8 on page 25in [BG06].

The product formula (1.1.1) enables us to define a height function on the algebraicpoints of projective space Pn: say K is still a number field and p = [p0 : · · · : pn] ∈ Pn(Q)with projective coordinates pi ∈ K, we set

H(p) =∏

v∈MK

max{|p0|v, . . . , |pn|v}dv/[K:Q].

Because of (1.1.1), another choice of algebraic projective coordinates for p leads to thesame height value. The product formula also implies H(p) ≥ 1.

If p ∈ Qn, then H(p) is often called the affine height and if p ∈ Pn(Q), then H(p)is called the projective height.

For notational purposes it is sometimes useful to take the logarithm. We define theabsolute logarithmic Weil height h(p) = log H(p) for p ∈ Qn or p ∈ Pn(Q).

If p = (p1, . . . , pn) ∈ Qn, then we have the useful estimates h(p) ≤ h(p1)+· · ·+h(pn)and max{h(p1), . . . , h(pn)} ≤ h(p) which follow from local considerations.

2. Height and Mahler measure of a polynomial

Keeping track of bounds of the size of a polynomial in integer coefficients is importantin transcendence theory when for example constructing auxiliary functions. “Size” couldmean for example the maximum of the absolute values of the coefficients or the sum overthese values. Of course more intricate definitions which anticipate a common divisor ofthe coefficients or which work if the coefficients are in a number fields are possible.

Let K be any field with an absolute value | · |. If f ∈ K[X1, . . . , Xn] is a polynomialwith coefficients fi1...in ∈ K, then we set

|f | = maxi1,...,in

{|fi1...in |}.

If | · | satisfies the ultrametric inequality and g ∈ K[X1, . . . , Xn], then by Gauss’s Lemmawe have |fg| = |f ||g|. In general equality does not hold if | · | does not satisfy theultrametric inequality.

Assume now that K is a number field and that f ∈ K[X1, . . . , Xn] is non-zero. Thenwe define the height of f as

hp(f) =1

[K : Q]

∑v∈MK

dv log |f |v.

Therefore the height of f is just the logarithmic absolute Weil height of the point inPn(Q) whose projective coordinates are just the non-zero coefficients of f . By (1.1.1)


and if λ ∈ Q∗ then hp(λf) = hp(f). We use the subscript p to distinguish hp(·) fromh(·) as there is danger of ambiguity if the polynomial in question is constant.

Let f ∈ C[X±11 , . . . , X±1

n ]\{0} and say | · | is now the complex absolute value. Thelogarithmic Mahler measure of f is given by

(1.2.1) m(f) =∫ 1

0· · ·∫ 1

0log |f(e2πit1 , . . . , e2πitn)|dt1 · · · dtn.

It is not immediately clear that this integral converges, as the absolute value in thelogarithm may vanish. For a proof of the existence of m(f) we refer to Lemma 2 page223 in [Sch00]. We also define M(f) = expm(f) and M(0) = 0. A direct and niceconsequence of (1.2.1) is the that M(fg) = M(f)M(g) for any f, g ∈ C[X±1

1 , . . . , X±1n ].

The Mahler measure M(f) can be bounded above and below by positive multiples of|f |. These factors depend only on the partial degrees of f .

If f ∈ C[X] is a polynomial in one variable with f = ad(X − α1) · · · (X − αd) andαi ∈ C, then by Jensen’s formula we have

(1.2.2) M(f) = |ad|max{1, |α1|} · · ·max{1, |αd|}.

Let τ ∈ Q have minimal polynomial f ∈ Q[X] with coprime integer coefficients, thenby (1.1.3) and (1.2.2) we have

(1.2.3) h(τ) =1

deg(f)m(f).

Hence heights and Mahler measures of integer polynomials are closed related.In general it is non-trivial to calculate the exact value of the Mahler measure of a

polynomial in more than one variable. Smyth’s work [Smy81] contains some explicitevaluations. For example he calculates the Mahler measure of the two variable polyno-mial X + Y − 1 in terms of an L-function. We will see more of this number in chapter3.

Even numerically approximating the integral in (1.2.1) can be tricky because of thepossible singularities in the integrand. In [Boy98] Boyd proposed an effective methodfor calculating M(f) up to arbitrary precision avoiding the approximation of an inte-gral. But he states that his method is impractical from a computational point of viewif f has more than one variable. Nevertheless Boyd’s general approach has an impor-tant theoretical consequence. The finite dimensional C-vector space of polynomials inC[X1, . . . , Xn] of degree bounded by some parameter carries a natural topology inducedby the topology on C. Boyd shows that for polynomials of bounded degree M(f) is acontinuous function of f . In the one-variable case this is a classical result going back toMahler.

In chapter 3 we will define a family of complex polynomials depending on a realparameter and study the function which maps this real parameter to the Mahler measureof the corresponding polynomial. More precisely, we show a uniform bound on thevariation of this function. Unfortunately Boyd or Mahler’s results on continuity do notsuffice in our situation.

1. A REVIEW OF HEIGHTS 15

We present two results on bounds for Mahler measures from Schinzel’s book [Sch00]which will be used in chapter 3.

We begin with an upper bound for the Mahler measure of a complex polynomialf in terms of the hermitian norm of the coefficient vector of P . Concretely, if P =fdX

d + · · ·+ f0 with fi ∈ C, then we set |f |2 = (|f0|2 + · · ·+ |fd|2)1/2.

Theorem 1.1 (Goncalvez). Let f = fdXd + · · ·+ f0 ∈ C[X] with f0fd 6= 0. Then

M(f)2 + |f0fd|2M(f)−2 ≤ |f |2with equality if and only if f(X)f(X−1) has just three non-zero coefficients.

Proof. This theorem is a special case of Theorem 40 in [Sch00]. �

The second result needed in chapter 3 is a lower bound for the Mahler measure ofpolynomials defined over certain number fields. We need a few definitions first.

Let f ∈ C[X]\{0}, then f is called self-inverse if there exists λ ∈ C∗ withXdeg ff(X−1) = λf(X).

Say now f ∈ K[X] where K is a number field. The content of f , denoted bycont(f), is the fractional ideal in K generated by the coefficients of f . For example if fhas coefficients in OK , then cont(f) is an ideal in OK .

We call a number field K Kroneckerian if K is a either totally real or a totallycomplex quadratic extension of a totally real number field.

We may now state Schinzel’s Theorem.

Theorem 1.2 (Schinzel). Let K be a Kroneckerian field and f ∈ K[X]\{0} notself-inverse with f(0) 6= 0. Then

(1.2.4)∏σ

M(σ(f)) ≥

(1 +

√5

2

)[K:Q]/2

N(cont(f)),

where the product runs over all embeddings σ : K → C.

Proof. This result is a special case of Theorem 72 in [Sch00]. Schinzel proves thistheorem for Laurent polynomials in any number of variables. �

The upper bound in Theorem 1.1 and the lower bound in 1.2 are so sharp that wewill use them to deduce irreducibility results for certain trinomials arising in chapter 3.

We note that Theorem 1.2 in the case K = Q implies a lower bound as in Smyth’spaper [Smy71] with a slightly worse constant. The important aspect of Schinzel’sTheorem is the exponent [K : Q] in (1.2.4).

CHAPTER 2

Multiplicative dependence and isolation I

In [CZ00] Cohen and Zannier proved that if x is algebraic with x and 1 − x mul-tiplicatively dependent, then max{H(x),H(1− x)} ≤ 2; here H(·) is the absolute Weilheight. This bound is sharp because of the exceptional values x = −1, 1/2, 2. Cohenand Zannier then used Bilu’s Equidistribution Theorem [Bil97] to prove an isolationresult: they showed that there exists ε > 0 such that if x is as before but not one ofthe three exceptional values, then max{H(x),H(1 − x)} ≤ 2 − ε. In this chapter wegive a concise proof of a slight strengthening of the height bound given in [CZ00]. Wealso work out an explicit ε. Finally we show that there exists a sequence xn with xn

and 1− xn multiplicatively dependent and such that the height of xn converges to theMahler measure of the polynomial X + Y − 1.

The contents of this chapter will appear in the Proceedings of the Pisa researchprogram “Diophantine Geometry” [Hab07].

1. Height bounds for dependent solutions of x + y = 1

Two elements x, y of a field are called multiplicatively dependent if xy 6= 0 and ifthere exist r, s ∈ Z not both zero such that xrys = 1.

We define M to be the set of complex x such that x and 1− x are multiplicativelydependent. Clearly the elements of M are algebraic. If ζ 6= 1 is a root of unity, thenζ and 1− ζ are multiplicatively dependent and so ζ ∈ M. Thus M is infinite, a resultwhich was made quantitative by Masser in Theorem 2 of [Mas07]. If y is also algebraicthen H(x, y) denotes the affine absolute non-logarithmic Weil height, which was definedin chapter 1. This height function corresponds to the compactification of the algebraictorus G2

m ↪→ P2. We have:

Theorem 2.1. Let x ∈ M, then H(x, 1 − x) ≤ 2 with equality if and only if x ∈{−1, 1/2, 2}.

Theorem 2.1 implies Theorem 1 of [CZ00] since max{H(x),H(y)} ≤ H(x, y) foralgebraic x and y. We choose the particular height function H(x, 1 − x) because it isinvariant under the maps x 7→ 1 − x and x 7→ x−1. Incidentally M is stable underthese two maps. Our method of proof for Theorem 2.1 exploits this fact and relies onelementary local estimates combined with the product formula. The proof of Theorem2.1 is a warm up for proof of Theorem 4.1 which gives a height bound for dependent x,y with x + y = α.

Theorem 2.1 makes explicit a special case of

17


Theorem ([BMZ99] page 1120). Let C be a closed absolutely irreducible curve inGn

m, n ≥ 2, defined over Q and not contained in a translate of a proper subtorus of Gnm.

Then the algebraic points of C which lie in the union of all proper algebraic subgroupsof Gn

m form a set of bounded Weil height.

Indeed x + y = 1 defines a line C in G2m which is not contained in a translate of

a proper subtorus. And any proper algebraic subgroup of G2m is contained in some

set defined by xrys = 1. Finally the Weil height used in [BMZ99] was the expressionH(x)H(y) ≥ H(x, y).

To prove the isolation result with explicit ε mentioned above we apply a result ofMignotte from [Mig89] on the angular distribution of conjugates of an algebraic numberof small height and big degree. Actually, large degree is guaranteed by a theorem ofSmyth [Smy71] on lower bounds for heights of non-reciprocal algebraic numbers.

Theorem 2.2. If x ∈M\{−1, 1/2, 2} then H(x, 1− x) < 1.915.

The element of M\{−1, 1/2, 2} of largest height known to the author is 1−ζ3 whereζ3 is a primitive 3rd root of unity. In fact H(1 − ζ3) =

√3. It would be interesting to

know if√

3 is already the second to largest height value obtained on M. In chapter 3we will develop an algorithm to decide this question.

If ζ 6= 1 is a root of unity, then 1−ζ ∈M. As the degree of ζ goes to infinity we canuse Bilu’s Equidistribution Theorem (Theorem 1.1, [Bil97]) to show that H(1− ζ, ζ) =H(1− ζ) converges to

(2.1.1) exp∫ 1/3

−1/3log |1 + exp(2πit)|dt = 1.381356...,

Let f be a polynomial in n variables with complex coefficients, the Mahler measureM(f) of f was defined in chapter 1 (1.2.1). Smyth ([Smy81]) calculated the Mahlermeasure of the polynomial X + Y − 1 as

(2.1.2) M(X + Y − 1) = exp(3√

34π

∑k≥1

(k

3

)1k2

),

here(··)

is the Legendre symbol. By Jensen’s formula the Mahler measure in (2.1.2) isequal to the integral (2.1.1). We immediately obtain:

Proposition 2.1. There exists a sequence xn ∈ M with limn→∞[Q(xn) : Q] = ∞such that limn→∞ H(xn, 1− xn) = M(X + Y − 1).

In chapter 3 we will prove a much stronger result. In fact H(xn, 1 − xn) convergesto M(X + Y − 1) for any sequence xn ∈M such that [Q(xn) : Q] goes to infinity.

2. Proof of Theorem 2.1

We prove Theorem 2.1 via an elementary estimate which holds for any field K withany absolute value | · | : K → R.

2. MULTIPLICATIVE DEPENDENCE AND ISOLATION I 19

Lemma 2.1. Let x ∈ K\{0, 1}, r, t ∈ Z with 0 6= t ≥ r ≥ 0 and xr = (1 − x)t. Wehave

(2.2.1) |1− x|−1 max{1, |x|} ≤ δ

where δ = 1 if | · | is ultrametric and δ = 2 otherwise. Furthermore, equality in (2.2.1)implies δ = 1 or r = 0 or r = t.

Proof. Let q denote the left-hand side of (2.2.1).First let us assume

(2.2.2) |x| < δ−1 or |x| > δ.

If δ = 1, then |x| 6= 1, so |1−x| = max{1, |x|}, hence q = 1. If δ = 2 we use the triangleinequality to bound

|1− x| ≥{|x| − 1 > |x|δ−1 : if |x| > δ,1− |x| > δ−1 : if |x| < δ−1

which implies q < δ. So in the case (2.2.2) we have q ≤ δ and furthermore q = δ canonly hold if δ = 1.

Now let us assume δ−1 ≤ |x| ≤ δ. If |x| < 1, then q = |1−x|−1 = |x|−r/t ≤ δr/t ≤ δ,and if |x| ≥ 1, then q = |x|/|1 − x| = |x|1−r/t ≤ δ1−r/t ≤ δ. It is clear that if we havethe equalities q = δ = 2, then r = 0 or r = t. �

We will see more arguments in this style in chapter 4.

Lemma 2.2. If ζ 6= 1 is a root of unity, then H(1 + ζ) ≤√

2√

3 = 1.8612...

Proof. Let K be a number field of degree d containing ζ. We multiply the productformula

∏v∈MK

|1− ζ|dvv = 1 with the definition of the height and note that ζ is an

algebraic integer to get

H(1 + ζ)d ≤ min{∏v|∞

max{1, |1 + ζ|v}dv ,∏v|∞

max{|1− ζ|v, |1− ζ2|v}dv}.

Let ∆1 be the set of infinite places v with |1 − ζ|v ≥ 1, let ∆2 be all other in-finite places. Recall that infinite places correspond to embeddings of K into C upto conjugation. If v ∈ ∆1, then elementary geometry gives |1 + ζ|v ≤

√3; with

the right-hand side replaced by 2 if we allow v ∈ ∆2. Similarly if v ∈ ∆2, thenmax{|1 − ζ|v, |1 − ζ2|v} ≤

√3; and if

√3 is replaced by 2, then the inequality holds

for v ∈ ∆1. We define δi =∑

v∈∆idv/d, then δ1 + δ2 = 1 and so

H(1 + ζ) ≤ min{√

3δ12δ2 , 2δ1

√3

δ2} =√

3(2√

3−1

)min{δ1,1−δ1} ≤√

2√

3.

�

We note that 12 log(2

√3) is an improvement of the trivial bound h(1 + ζ) ≤ log 2

which holds for any root of unity ζ. In the proof of Theorem 2.1 we need only a weakform of Lemma 2.2, namely the fact that H(1 + ζ) < 2 is ζ 6= 1 is a root of unity.


Lemma 2.3. Let x′ ∈ M, then there exist x ∈ M and r, t ∈ Z with 0 6= t ≥ 2r ≥ 0such that xr = (1− x)t and h(x′, 1− x′) = h(x). Furthermore, if x′ /∈ {−1, 1/2, 2} thenwe can choose x such that x /∈ {−1, 1/2, 2}.

Proof. The lemma is simple if x′ is a primitive 6th root of unity, for then 1 − x′

is also a 6th root of unity and we may take x = x′, t = 6, and r = 0. Hence it sufficesto show the lemma for x′ ∈M0 with M0 = M\{e±2πi/6}. For any such x′ there existsa unique λ(x′) = [r : t] ∈ P1(Q) with r and t coprime integers such that x′r(1 − x′)−t

is a root of unity. The maps φ1(x) = 1/x and φ2(x) = 1 − x are automorphisms ofM0 and generate the symmetric group S3. Thus we get an action of S3 on M0 whichalso leaves {−1, 1/2, 2} invariant. By the product formula the height h(x, 1− x) is alsoinvariant under this action. We check that if λ(x) = [r : t], then λ(φ1(x)) = [t − r : t]and λ(φ2(x)) = [t : r]. We get an action of S3 on P1(R). Furthermore, any element ofP1(R) lies in the orbit of some element of {[1 : s]; s ≥ 2}∪{[0 : 1]}. The lemma followssince if xr(1− x)−t is a root of unity with t ≥ 2r ≥ 0, then h(x, 1− x) = h(x). �

Proof of Theorem 2.1: Because of Lemma 2.3 it suffices to show that if x ∈ Q\{0, 1}with x 6= −1, 1/2, 2 and xr = (1− x)t for integers 0 6= t ≥ 2r ≥ 0, then h(x) < log 2.

If r = 0, then x = 1 + ζ for some root of unity ζ 6= ±1. In this case the theoremfollows from Lemma 2.2.

Let us assume r > 0. We fix a number field K that contains x and apply the productformula (1.1.1) to 1− x to deduce

[K : Q]h(x) =∑

v

dv log max{1, |x|v} =∑

v

dv logmax{1, |x|v}|1− x|v

,

where dv are the local degrees from chapter 1. Since 0 < r < t we apply Lemma 2.1 tothe local terms in the equality above to see that [K : Q]h(x) <

∑v infinite[Kv : Qv] log 2.

This inequality completes the proof since the sum is just [K : Q] log 2. �


A non-zero algebraic number α is called reciprocal if α and α−1 are conjugated.We apply Mignotte’s equidistribution result and Smyth’s Theorem ([Smy71]) on lowerbounds for heights of non-reciprocal algebraic integers to deduce the following lemma.

Lemma 2.4. Let α ∈ Q∗ be non-reciprocal with h(α) ≤ log 23·105 , then h(1 + α) ≤

0.933 · log 2 + h(α).

Proof. Let α be as in the hypothesis and d = [Q(α) : Q], furthermore let θ0 > 1be the unique real that satisfies θ3

0 − θ0 − 1 = 0. If α is an algebraic integer, thendh(α) ≥ log θ0 by Smyth’s Theorem ([Smy71]). The upper bound for h(α) implies

(2.3.1) d ≥ 121700.

On the other hand, if α is not an algebraic integer, then it is well-known that dh(α) ≥log 2. Thus (2.3.1) holds in any case.

2. MULTIPLICATIVE DEPENDENCE AND ISOLATION I 21

We split C∗ up into three sectors

Ck = {r · exp(iφ); r > 0 and2π

3(k − 1) ≤ φ <

2π

3k} for 1 ≤ k ≤ 3

and define the function

m(z) =max{1, |z + 1|}

max{1, |z|}=

max{1, (r2 + 2r cos φ + 1)1/2}max{1, r}

for z = r · exp(iφ) with r > 0 and φ ∈ R. Hence

m(z)2 ≤

{max{1,r2+2r+1}

max{1,r2} : if −2π/3 ≤φ ≤ 2π/3max{1,r2−r+1}

max{1,r2} : if 2π/3 ≤φ ≤ 4π/3.

Elementary calculus now leads to

(2.3.2) m|C1∪C3 ≤ 2 and m|C2 ≤ 1.

We fix an embedding of Q into C. Let α1. . . . , αd ∈ C∗ be the conjugates of α.We set Nk = |{i; αi ∈ Ck}| for 1 ≤ k ≤ 3. For any finite place v of Q(α) we havemax{1, |1 + α|v} = max{1, |α|v} by the ultrametric inequality. Since the infinite placesof Q(α) taken with multiplicities correspond to embeddings of Q(α) into C and becauseof (2.3.2) we have

(2.3.3) d(h(1 + α)− h(α)) =d∑

i=1

log m(αi) ≤ (N1 + N3) log 2.

We set ε = (92c2( log(2d+1)

d + h(α)))1/3 with c = 2.62. Since log(2d+1)d is decreasing

considered as function of d ≥ 1, we use (2.3.1) and our hypothesis on h(α) to concludeε < 0.1477. We apply Mignotte’s Theorem (ii) ([Mig89], page 83) to the minimalpolynomial of α and to the closure of our sectors Ck to bound

(2.3.4)Nk

d≤ 1

3+ 2.823(

log(2d + 1)d

+ h(α))1/3.

Our hypothesis on h(α) and (2.3.1) together with (2.3.4) imply Nkd < 0.4662. This last

bound applied to (2.3.3) concludes the proof. �

We note that the trivial bound h(1+α) ≤ log 2+h(α) holds for all algebraic α. ThusLemma 2.4 gives a slight improvement for non-reciprocal α of small height. Instead ofSmyth’s lower bound for heights we could have used the lower bound by Dobrowolskiwhich holds for any non-zero algebraic number not a root of unity. This approachleads to slightly worse numerical constants. By taking sectors with smaller angles inthe proof of Lemma 2.4 the constant 0.933 · log 2 can be replaced by any real numberstrictly greater than the logarithm of the number (2.1.1) if the height of α is sufficientlysmall but positive. But the bound given in Lemma 2.4 is apt for our application.


In [CZ00] Cohen and Zannier introduced a function S : (1,∞) → R relevant to ourproblem. We briefly recall its definition. Say λ > 1 and let ξ, ξ > 1 be the unique realssuch that ξλ = ξ + 1 and ξλ/(λ−1) = ξ + 1, then

S(λ) =log(ξ + 1) log(ξ + 1)

log(ξ + 1) + log(ξ + 1).

Lemma 1 of [CZ00] implies S < log 2, furthermore if xr = (1 − x)t for integerst > r > 0, then h(x) ≤ S(t/r). The proof of said lemma also shows that S increases on[2,∞).Proof of Theorem 2.2: Because of Lemma 2.3 it suffices to show that if x ∈ Q\{0, 1},x 6= −1, 1/2, 2 and xr = (1 − x)t for integers 0 6= t ≥ 2r ≥ 0, then h(x) < log 1.915. Ifr = 0, then x = 1+ ζ for a root of unity ζ 6= 1. Lemma 2.2 implies h(x) ≤ 1

2 log(2√

3) <log 1.915. We now assume r > 0 and define λ = t/r ≥ 2.

If λ < 3 ·105, then we have h(x) ≤ S(3 ·105) by the properties of S(·). A calculationshows that the right-hand side of the last inequality is strictly less then log 1.915.

Finally we assume λ ≥ 3 · 105. Then h(1 − x) = λ−1h(x) ≤ log 23·105 by Theorem 2.1.

Let α = x− 1, we have

(2.3.5) (−1)tαt = (1 + α)r.

Let us assume first that α and α−1 are not conjugated, then h(x) ≤ 0.933 · log 2+ log 23·105 <

log 1.915 by Lemma 2.4. If α and α−1 are conjugated, then equality (2.3.5) must holdwith α replaced by α−1. Hence 1 = α2t(1+α−1)2r, or 1 = x2r(1−x)2(t−r) in terms of x.Since r 6= 0 and r 6= 2t this new dependency relation between x and 1−x is independentof the original relation 1 = xr(1 − x)−t. We conclude that x is a root of unity and soh(x) = 0. �

CHAPTER 3

Multiplicative dependence and isolation II

In this chapter we continue the study of multiplicatively dependent solutions x, yof the equation x + y = 1. We recall that M is the set of x ∈ Q\{0, 1} such that xand 1 − x are multiplicatively dependent. It was first proved by Cohen and Zannierin [CZ00] that if x ∈ M, then max{h(x), h(1 − x)} ≤ log 2 with equality if and onlyif x ∈ {−1, 1/2, 2}. Furthermore, they showed that log 2 is an isolated point in therange of this particular height function restricted to M. In chapter 2 we worked withthe possibly larger height h(x, 1 − x) and also got log 2 as an upper bound. We alsoshowed that if x ∈ M\{−1, 1/2, 2}, then h(x, 1 − x) ≤ log 1.915, thus making Cohenand Zannier’s isolation result explicit. It is thus natural to ask for the exact value ofthe greatest limit point of the height. We will answer this question and even determineall limit points.

1. Introduction

LetS0 = m(X + Y − 1) = 0.3230659472194505140936 . . .

where m(·) is the logarithmic Mahler measure of a polynomial in complex coefficients,see (1.2.1). One may interpret S0 as the logarithmic height of the curve defined byX + Y = 1 in G2

m. In [Smy81] Smyth expressed S0 in terms of the value of the L-function associated to χ, the (unique) non-trivial character of conductor 3, at s = 2, cf.chapter 2 equation (2.1.2).

In this chapter we prove that the set of height values

Σ = {h(x, 1− x); x ∈M}

has exactly one limit point equal to S0. It follows that all points in Σ with at mostone exception are isolated. We cannot discard of this one exception because we cannotexclude that S0 is itself a height value. In fact it seems to be already unknown if S0 isthe logarithm of an algebraic number or not.

We now state the Limit Point Theorem:

Theorem 3.1. Let ε > 0 there is c(ε) depending only on ε such that if x ∈ M andD = [Q(x) : Q] then

|h(x, 1− x)− S0| ≤c(ε)D

e(ε+log 2) log D

log log 3D .

23


The constant c(ε) in Theorem 3.1 is effective, see Proposition 3.1 for an explicitbound of |h(x, 1− x)− S0|.

Of course the theorem implies the boundedness of h(x, 1−x) for x ∈M, but it doesnot give the optimal bound log 2.

We deduce two immediate consequences of the previous theorem and Northcott’sTheorem.

Corollary 3.1. Let xn ∈ M be a sequence such that each xn occurs only finitelyoften, then h(xn, 1− xn) → S0 as n →∞.

Corollary 3.2. The set {h(x, 1− x); x ∈M}\{S0} ⊂ R is discrete.

It seems natural to conjecture that the set {h(x, 1− x); x ∈M} ⊂ R is discrete.The essential tool used in the proof of the Limit Point Theorem is the following

irreducibility result on certain trinomials which may be of independent interest. Werecall that a number field is called Kroneckerian if it is either totally real or a totallycomplex quadratic extension of a totally real number field.

Theorem 3.2. Assume K is a Kroneckerian number field. Let n ≥ m ≥ 0 becoprime integers, ξ, ζ roots of unity in K, and P = Xn + ξXm + ζ ∈ K[X]. LetP = AB where A,B ∈ K[X], A /∈ K, and such that the zeros of B are precisely theroots of unity that are zeros of P . Then deg B ≤ 2, furthermore A is irreducible inK[X] except if P = X2 + ξX − ξ2 and

√5 ∈ K.

This theorem may be compared with results obtained independently by Ljunggrenin [Lju60] and by Tverberg in [Tve60]. They give a complete factorization over Q oftrinomials such as in Theorem 3.2 with ξ, ζ ∈ {±1}.

A result related to Theorem 3.2 was proved by Schinzel in Corollary 8 ([Sch00] page417). As in his proof we prove the irreducibility of A by showing that if g is the numberof irreducible factors of A, then g < 2. This bound for g follows from comparing upperand lower bounds for the Mahler measure of polynomials with coefficients in K. Theupper bound is due to Goncalvez (Theorem 1.1) and the lower bound is Theorem 1.2showed by Schinzel. Our proof of Theorem 3.2 follows Schinzel’s proof of Corollary 8.

Because X2+ξX−ξ2 = (X−ξα)(X−ξβ) for α = (−1+√

5)/2, β = (−1−√

5)/2, thispolynomial is reducible over Q(ξ) precisely when this field contains

√5. Furthermore,

this situation does occur for a root of unity ξ. Indeed since Q(√

5) is abelian extensionof Q it is contained in a field generated by a root of unity by the Theorem of Kronecker-Weber. Or by an elementary argument: if ξ is a primitive 5th root of unity, then it iswell-known that (ξ − ξ2 − ξ3 + ξ4)2 = 5.

For example Theorem 3.2 implies that X2007 +X +1 is irreducible in Qab[X], whereQab is the maximal abelian extension of Q.

We give a short description of the proof of Theorem 3.1: Theorem 3.2 enables usto express the height of essentially all x ∈ M in terms of the average over the Mahlermeasure of the conjugates of a certain trinomial much like the one in Theorem 3.2. Weuse Koksma’s inequality to compare this average with a 2-dimensional integral which

3. MULTIPLICATIVE DEPENDENCE AND ISOLATION II 25

equals S0. Two steps are required to get an explicit bound in Koksma’s inequality. Firstwe apply a classical result of Erdos and Turan to show that roots of unity of fixed orderare sufficiently well distributed around the unit circle. Second we need a uniform boundfor the total variation of a family of functions related to Mahler measures of certaintrinomials. The key estimate in bounding the total variation will be Lemma 3.13 whichfollows from a lengthy calculation.

We remark that it is not completely evident that the set of height values Σ is aninfinite set. The infinitude of Σ can be proved by an argument kindly mentioned to meby Masser and Zannier. In fact if ζ is a primitive pth root of unity and ξ is a primitiveqth root of unity with p and q primes, then h(1−ζ) = h(1−ξ) if and only if p = q. Indeedlet us assume p 6= q and H(1− ζ) = H(1− ξ) where H(·) = exph(·). By the definitionof the height it is clear that H(1 − ζ)2(p−1)(q−1) = H(1 − ξ)2(p−1)(q−1) ∈ Q(ζ) ∩Q(ξ).These two fields have intersection Q, hence H(1 − ζ)2(p−1)(q−1) is a rational and evenan integer. Since any conjugate of 1 − ζ generates the unique prime ideal above p inthe ring integers of Q(ζ) we conclude that H(1− ζ)2(p−1)(q−1) is a power of p. SimilarlyH(1− ζ)2(p−1)(q−1) is a power of q. An easy calculation shows that 1 − ζ is not a rootof unity (cf. the remark at the beginning of section 2). So by Kronecker’s TheoremH(1− ζ) > 0 and we have a contradiction.

On the other hand, carefully calculating the height shows that h(1 + ζ) < S0 <h(1 − ζ) if ζ is a primitive pth root of unity for any prime p. We will not show thisinequality here. Together with Theorem 3.1 it gives another proof of the fact that Σ isinfinite. Moreover, we conclude that h(x, 1− x) attains a second, third, etc. maximumrespectively minimum as x runs over M. Since Theorem 3.1 is effective, one could usea machine to calculate these values.

In [Zag93] Zagier asked the following question on the set of small height valuesh(x) + h(1 − x) where x ∈ Q is now unrestricted: does there exist a sequence c1 <c2 < · · · such that h(x) + h(1 − x) = ci has finitely many algebraic solutions andh(x) + h(1− x) > lim sup ci for all other x ∈ Q? In this more general setting no answeris known.

The height used by Cohen and Zannier was max{h(x), h(1 − x)}. With Theorem3.1 we can determine the limit points of Σ′ = {max{h(x), h(1− x)}; x ∈M}:

Corollary 3.3. The closure of Σ′ ⊂ R is equal to the union of [S0/2, S0] with adiscrete set.

We complete this section by giving a further application to Theorem 3.2. The subsetof elements in M with degree bounded by a parameter is finite by Northcott’s Theoremand Theorem 2.1. In [Mas07] Masser proposed the problem of counting elements in Mof bounded degree. More precisely, given a real D and

M(D) = {x ∈M; [Q(x) : Q] ≤ D},

can one say something about the asymptotic behavior of the cardinality |M(D)| as Dgoes to infinity?


Let φ be Euler’s totient function and D ≥ 3. Masser proved the bounds

(3.1.1) c0D3 − 200D2(log D)3 ≤ |M(D)| ≤ 1018D3 (log D)9

(log log D)6

with

c0 =6π2

∞∑k=1

1φ(k)2

.

Masser obtained the lower bound in (3.1.1) by carefully counting zeros of trinomials.The upper bound comes from an explicit “relative Lehmer”-type height lower boundproved by Pontreau in [Pon05]. The intricate definition of c0 has some significance: in[Mas07] Masser conjectured

(3.1.2) limD→∞

|M(D)|D3

= c0.

With the help of Theorem 3.2 we will prove Masser’s conjecture by improving theupper bound in (3.1.1). We even give an asymptotic equality.

Theorem 3.3. Let D ≥ 3, then

(3.1.3) c0D3 − 200D2(log D)3 ≤ |M(D)| ≤ c0D

3 + 1000D2(log D)3.

In particular (3.1.2) holds.

2. Factorizing trinomials over Kroneckerian fields

Let S1 ⊂ C be the unit circle around 0. We will often use the elementary fact thatif x+y = 1 with x, y ∈ S1, then x and y are primitive sixth roots of unity with x = y−1.In particular there are only two possibilities for (x, y).

Lemma 3.1. Let n > m > 0 be integers, ξ, ζ ∈ S1, and P = Xn + ξXm + ζ ∈ C[X].Then P has only simple zeros. Furthermore, if z ∈ C∗ with P (z) = P (z−1) = 0, thenzn−2m = ξ2ζ−1.

Proof. We begin by proving the first assertion in the lemma. Let z be a zero of Pwith multiplicity > 1. In this case we have zn + ξzm = −ζ and nzn + ξmzm = 0. Weconsider the two previous equalities as linear equations in unknowns zn and zm. Sincen 6= m we can solve for zn and zm:

zn =m

n−mζ, zm = − n

n−m

ζ

ξ.

Taking absolute values and setting µ = m/n gives |zn| = µ/(1−µ) and |zm| = 1/(1−µ).Hence µµ(1− µ)1−µ = 1, a contradiction since 0 < µ < 1.

The second part of the assertion is very easy. Let z ∈ C∗ with

(3.2.1) zn + ξzm + ζ = 0 and z−n + ξz−m + ζ = 0.

So also zn + ζξzn−m + ζ = 0. We subtract this equality from the first one in (3.2.1) tosee that zn−2m = ξ2ζ−1. �


Proof of Theorem 3.2: If n = m or m = 0, then the result is immediate as in this casewe have n = 1. So let us also assume n > m > 0, in particular P (0) 6= 0.

Let P = AB be a decomposition as in the statement of Theorem 3.2; we may assumethat A and B are monic. Say B(η) = 0, then η is a root of unity and P (η) = 0. Hence−ζ−1ηn + (−ζ−1ξηm) = 1. We conclude that −ζ−1ηn = x, −ζ−1ξηm = x−1 where xis a primitive 6th root of unity. Since n and m are coprime there exist a, b ∈ Z witham + bn = 1. Hence xb−a = (−1)a+bζ−a−bξaη. Since there are only 2 possibilities forx, there are at most 2 such η. By Lemma 3.1 we conclude deg B ≤ 2. The first part ofthe assertion follows.

Now let A = A1 · · ·Ag where the Ai ∈ K[X] are irreducible and monic. By hypoth-esis A /∈ K, so g ≥ 1.

We claim that the Ai are not self-inverse, i.e. Xdeg AiAi(X−1)Ai(X)−1 is not aconstant. Indeed let us assume the contrary. If Ai(z) = 0, then z 6= 0 and P (z) =P (z−1) = 0. By Lemma 3.1 we have zn−2m = ξ2ζ−1. Since z is not a root of unity wehave n = 2m, so n = 2 and m = 1. Hence ξ2 = ζ and so P = X2 + ξX + ξ2. But nowall zeros of P are roots of unity, contradicting our assumption A /∈ K. Therefore noneof the Ai are self-inverse.

By Gauss’s Lemma we have Ai ∈ OK [X], and since the Ai are monic we haveN(cont(Ai)) = 1. By Theorem 1.2 we get the lower bound

∏σ

M(σ(Ai)) ≥

(√5 + 12

)[K:Q]/2

where the product runs over all embeddings σ : K → C extended to K[X] in theusual manner. The Mahler measure is multiplicative and we have M(σ(B)) = 1 byconstruction, therefore

∏σ

M(σ(P )) =g∏

i=1

∏σ

M(σ(Ai)) ≥

(√5 + 12

)g[K:Q]/2

.(3.2.2)

Let us assume for the moment that P (X)P (X−1) has strictly more than 3 non-zeroterms. Let σ : K → C be an embedding, we will bound M(σ(P )) from above. Sinceσ(P ) has three coefficients of modulus 1 Theorem 1.1 in chapter 1 implies M(σ(P )) <√

5+12 . If we apply this last inequality to bound the left side of (3.2.2) we conclude g < 2,

hence A = A1 is irreducible.But what if

P (X)P (X−1)

= ζ−1Xn + ζX−n + ξ−1Xn−m + ξXm−n + ξζ−1Xm + ξ−1ζX−m + 3

has at most 3 non-zero terms? Since n > m > 0 are coprime we must have n = 2,m = 1, and ξ2 = −ζ. Therefore P = X2 + ξX − ξ2, and this polynomial has no rootsof unity as zeros, so P = A. As already pointed out in section 1 the polynomial A is


reducible K if and only if√

5 ∈ K, but this is just the case we are excluding in theassertion. �

3. A reduction

We define a norm || · || on R2 which will be used in certain places of this chapter.Let (r, s) ∈ R2, we set

||(r, s)|| ={|r|+ |s| if rs ≥ 0,max{|r|, |s|} if rs < 0.

The following lemma will be helpful throughout the whole chapter, it is similar toLemma 2.3.

Lemma 3.2. Let x ∈M. There exist a root of unity ζ, integers k, l,m, n, and α ∈ Qsuch that

n ≥ 2m ≥ 0, (n−m)k − nl = 1, h(x, 1− x) = nh(α),

αn−ζ lαm + ζk = 0, and Q(x) = Q(α, ζ).

Furthermore, there exist coprime integers r and s such that xr(1 − x)s = ±ζ and||(r, s)|| = n.

Proof. The lemma is simple if x and 1− x are both roots of unity. Indeed in thiscase x is a primitive 6th root of unity and x + x−1 = 1. We set α = x, ζ = x−1, n = 1,m = 0, k = 1, l = 0, r = 0, and s = 1.

Let M0 = M\{e±2πi/6}, then for any x ∈M0 there is a unique λ(x) ∈ P1(Q) suchthat if λ(x) = [r : s] with r, s ∈ Z coprime then xr(1− x)s is a root of unity. Since thepair (r, s) is uniquely determined up to sign it makes sense to speak of ||λ(x)||. Thetwo maps φi : M0 →M0 given by φ1(x) = 1/x, φ2(x) = 1− x generate the symmetricgroup S3 which acts on M0. These actions leave h(x, 1 − x) invariant by the productformula. Furthermore, we have Q(φi(x)) = Q(x). We check that if λ(x) = [r : s] withcoprime r, s ∈ Z, then λ(φ1(x)) = [r + s : −s] and λ(φ2(x)) = [s : r]. So we have anaction of S3 on P1(R). Let w ∈ R\{−1, 0}, then the orbit of [1 : w] equals{

[1 : w], [1 : − w

1 + w], [1 : −1− 1

w], [1 : −1− w], [1 : − 1

1 + w], [1 :

1w

]}

.

It is easily verified that

infw∈R\{−1,0}

max{w,− w

1 + w,−1− 1

w,−1− w,− 1

1 + w,

1w} ≥ 1.

Since the orbit of [0 : 1] contains [1 : −1] and [1 : 0] we conclude that any point inP1(R) is in the orbit of some element of F = {[1 : w]; w ≥ 1 in R} ∪ {[0 : 1]}. Finallyit is simple to show ||λ(x)|| = ||λ(φi(x))||.

Let x ∈ M0, by letting S3 act on x we assume λ(x) = [r : s] ∈ F with r, s ∈ Zcoprime and r ≥ 0, therefore s ≥ r and s > 0. So xr(1− x)s = ζ for some root of unityζ. We choose integers a, b with ar + bs = 1 and set α = xb(1− x)−a. Then αs = xζ−a


and α−r = ζ−b(1− x). Clearly we have Q(x) = Q(α, ζ) and αr+s − ζ−aαr + ζb−a = 0.We choose n = r+s, m = r, k = b−a, and l = −a. By definition ||(r, s)|| = r+s = n. Itremains to show the height equality in the assertion. We note h(x, 1−x) = h(x−1, x−1−1) by the product formula. We have ζx−r−s = (x−1 − 1)s, so by local considerationsh(x−1, x−1 − 1) = h(x−1 − 1) = (1 + r/s)h(x) = (r + s)h(α) = nh(α). �

We apply Theorem 3.2 to study irreducible factors of the trinomial in the previouslemma. If ζ is a root of unity we let ord ζ denote its order.

Lemma 3.3. Let x ∈ M and h(x, 1 − x) > 0. In the situation of Lemma 3.2 setK = Q(ζ) and R = Xn − ζ lXm + ζk. There exist A,B ∈ K[X] with A irreducible,the zeros of B are precisely the roots of unity that are zeros of P , and deg B ≤ 2.Furthermore, if ord ζ - 6 then B ∈ K.

Proof. Let R = AB where the zeros of B are precisely the roots of unity that arezeros of R. We would like to apply Theorem 3.2. We first note that A /∈ K since A 6= 0and A(α) = 0 where α is no root of unity. Next we dismiss the case where R is oneof the exceptions described in the statement of Theorem 3.2. What happens if n = 2,m = 1, and ζ2l = −ζk? Since (n−m)k− nl = 1 we conclude k− 2l = 1 and so ζ = −1.But then K = Q. Theorem 3.2 applies in our situation since

√5 /∈ Q. Hence deg B ≤ 2

and A ∈ K[X] is irreducible.Let us assume that R has a zero η which is a root of unity. Then −ζ−kηn+ζ l−kηm =

1. Hence (ζ−kηn)6 = (ζ l−kηm)6 = 1. So

1 = (ζ−kηn)−6m(ζ l−kηm)6n = ζ6(km+ln−kn) = ζ−6.

Therefore if ord ζ - 6, then B has no zeros, hence B ∈ K. �

4. Averaging the Mahler measure

Let θ ∈ R and let n ≥ m ≥ 0 be integers with n 6= 0. We introduce the followingnotation

P (X, θ) = Pnm(X, θ) = Xn − e−2πiθXm + 1,

Q(X, θ) = Qnm(X, θ) = (Xn − e−2πiθXm + 1)(X−n − e2πiθX−m + 1)(3.4.1)

f(θ) = fnm(θ) =∫ 1

0log |Pnm(e2πiu, θ)|du.

Clearly f satisfies f(θ + 1) = f(θ). This function equals log M(Pnm(X, θ)) andis therefore continuous by a classical result of Mahler. If ζ ∈ S1, then Qnm(ζ, θ) =|Pnm(ζ, θ)|2 ≥ 0. So for example

(3.4.2) fnm(θ) =12

∫ 1

0log Qnm(e2πiu, θ)du.

We will often omit the indices n and m to ease notation.


Lemma 3.4. Let a, d be positive integers and ζ = e2πia/d, furthermore let k, l,m, nbe integers with n ≥ m ≥ 0 and (n−m)k − nl = 1. Then

log M(Xn − ζ lXm + ζk) = fnm(a

dn).

Proof. We have

log M(Xn − ζ lXm + ζk) =∫ 1

0log |e2πi(nu− ka

d) − e2πi(mu+a

d(l−k)) + 1|du.

Of course n > 0, so the change of variables u′ = u− akdn leads to

log M(Xn − ζ lXm + ζk) =∫ 1− ak

dn

− akdn

log |e2πinu′ − e2πi(mu′− adn

) + 1|du′.

Since the integrand above is invariant under u′ 7→ u′ +1 we may replace the integrationlimits by 0 and 1. This completes the proof. �

Lemma 3.5. Let x ∈ M and h(x, 1 − x) > 0. In the situation of Lemma 3.2 andwith d = ord ζ we have

(3.4.3) h(x, 1− x) =n

[Q(ζ, α) : Q]

∑1≤a≤d, (a,d)=1

fnm(a

dn).

Furthermore, n− 2 ≤ [Q(ζ, α) : Q(ζ)] ≤ n in general and [Q(ζ, α) : Q(ζ)] = n if d - 6.Finally φ(d) ≤ [Q(x) : Q].

Proof. By Lemma 3.2 it suffices to show that nh(α) is equal to the right side of(3.4.3). Let R be the polynomial given implicitly in the assertion of Lemma 3.2, i.e.R = Xn − ζ lXm + ζk ∈ K[X] with K = Q(ζ); we have R(α) = 0. Also n ≥ 2m and son > m since n and m cannot be both zero. In particular α is an algebraic integer.

By Lemma 3.3 we may factor R = AB with A,B ∈ K[X], where A is irreducibleand monic, and such that the zeros of B are precisely the roots of unity that are zeros ofR. We note that B is also monic. Since α is not a root of unity we conclude A(α) = 0.Therefore A is the minimal polynomial of α over K. So A′ =

∏σ σ(A) is the minimal

polynomial of α over Q, here σ runs over all embeddings K → C. Since A′ is monicand α is an algebraic integer we conclude A′ ∈ Z[X] and therefore by (1.2.3)

(3.4.4) h(α) =1

[Q(x) : Q]log M(A′) =

1[Q(x) : Q]

∑σ

log M(σ(A)).

The zeros of the monic polynomial B are roots of unity and the Mahler measure ismultiplicative, so we may replace A by R in (3.4.4).

Equality (3.4.3) follows immediately from (3.4.4), Lemma 3.4, and Q(x) = Q(ζ, α).The assertions regarding [Q(ζ, α) : Q(ζ)] follow from Lemma 3.3. Since φ(d) = [Q(ζ) :Q] and Q(ζ) ⊂ Q(x) we conclude φ(d) ≤ [Q(x) : Q]. �


5. Bounding the variation

Throughout this section we use the notation from Lemma 3.2, i.e. n ≥ 2m ≥ 0 arecoprime integers and (n−m)k − nl = 1 for integers k, l. These properties imply n > 0so we may define µ = m/n ∈ [0, 1/2]. Furthermore, we keep the notation introducedaround (3.4.1)

We define E = Enm to be the set of all θ ∈ [0, 1] such that Qnm(ζ, θ) = 0 for someζ ∈ S1. Equivalently, Enm is the set of θ ∈ [0, 1] with Pnm(ζ, θ) = 0 for some ζ ∈ S1.

Lemma 3.6. The set Enm is finite.

Proof. We assume that ζ ∈ S1 satisfies P (ζ, θ) = 0. Then −ζn + e−2πiθζm = 1and so

ζ6n = (e−2πiθζm)6 = 1.

We divide the mth power of the first expression by the nth power of the second expressionand conclude e12πiθn = 1. Since n 6= 0 there are only finitely many θ ∈ [0, 1] that satisfythis last inequality. �

Lemma 3.7. Let θ ∈ [0, 1]\Enm and say m > 0. We consider Qnm(X, θ) as aLaurent polynomial in X. If z 6= 0 is a zero of Qnm(X, θ), then it is a simple zero, andfurthermore ∂

∂θQnm(z, θ) 6= 0.

Proof. For brevity we set ξ = e−2πiθ with θ ∈ [0, 1]\E. By (3.4.1) we have

Q(X, θ) = P (X, θ)P (X−1,−θ),(3.5.1)∂

∂XQ(X, θ) =

∂P

∂X(X, θ)P (X−1,−θ)− P (X, θ)

1X2

∂P

∂X(X−1,−θ).(3.5.2)

We prove the first part by contradiction. So assume z 6= 0 is a zero of Q(X, θ) ofmultiplicity > 1, hence z is a zero of both (3.5.1) and (3.5.2). By our hypothesis on θwe have |z| 6= 1.

Let us assume first that P (z, θ) = 0. By Lemma 3.1 ∂∂X P (X, θ) is non-zero at z.

We get P (z−1,−θ) = 0 by (3.5.2). We apply the second statement in Lemma 3.1 and|z| 6= 1 to conclude that n = 2m and ξ2 = 1. Thus z2m− ξzm + ξ2 = 0, hence zm/ξ is aroot of unity, a contradiction.

The case P (z−1,−θ) = 0 is similar and also leads to a contradiction.We turn to the second part of the lemma. We expand Q and its derivative:

Q = Xn + X−n − ξ−1Xn−m − ξXm−n − ξXm − ξ−1X−m + 3,(3.5.3)∂Q

∂θ= 2πi(ξXm−n − ξ−1Xn−m + ξXm − ξ−1X−m).(3.5.4)

If z 6= 0 is a common zero of Q(X, θ) and ∂∂θQ(X, θ), we will derive a contradiction.

We define a = zn and b = −ξzm. Then by (3.5.3) and (3.5.4) we have the identitiesa + a−1 + ab−1 + a−1b + b + b−1 + 3 = 0 and ab−1− a−1b− b + b−1 = 0. We add the twoto obtain

(3.5.5) a + a−1 + 2ab−1 + 2b−1 + 3 = 0.


By (3.4.1) we have either a + b + 1 = 0 or a−1 + b−1 + 1 = 0. In both cases we can use(3.5.5) to deduce that a is a root of unity. Therefore so is z, this contradicts our choiceθ /∈ E. �

We define

Tnm(X, θ) = (∂Qnm

∂θ/Qnm)(X, θ).

Lemma 3.8. Let θ ∈ [0, 1]\Enm and say m > 0. If Pnm(z, θ) = 0 then z 6= 0,1 + z−n 6= 0, µ− (1 + z−n)−1 6= 0, and the residue satisfies

resX=zTnm(X, θ)

X= −2πi

n

(µ− 1

1 + z−n

)−1

.

Proof. Say P (z, θ) = 0, then z 6= 0 since n > m > 0. We set a = zn, ξ = e−2πiθ,and b = −ξzm, hence a + b = −1. The second non-vanishing statement follows since1 + z−n = 1 + a−1 = −ba−1.

Lemma 3.7 implies that the non-zero poles of T (X, θ)/X are precisely the zeros ofQ(X, θ) and that these poles are all simple. Thus the residue of T (X, θ)/X at z is givenby

resX=zT (X, θ)

X=

1z

∂∂θQ(X, θ)∂

∂X Q(X, θ).

With this equality, (3.5.3), and (3.5.4) we evaluate

resX=zT (X, θ)

X(3.5.6)

= −2πia−1b− ab−1 + b− b−1

na− na−1 + (n−m)ab−1 − (n−m)a−1b + mb−mb−1

= −2πi

n

b2 − a2 + ab2 − a

a2b− b + a2 − a2µ− b2 + b2µ + ab2µ− aµ

= −2πi

n

(a + 1)(a2 + a + 1)(µ(a + 1)− a)(a2 + a + 1)

= −2πi

n

(µ− 1

1 + a−1

)−1

,

here the second to last equality follows by using a + b = −1. This calculation alsoimplies the last non-vanishing statement in the assertion. �

The next Lemma is very similar to Lemma 3.8

Lemma 3.9. Let θ ∈ [0, 1]\Enm and say m > 0. If Pnm(z, θ) = 0 then z 6= 0,1 + z−n 6= 0, µ− (1 + z−n)−1 6= 0, and the residue satisfies

resX=z−1

Tnm(X, θ)X

= −2πi

n

(µ− 1

1 + z−n

)−1

.


Proof. The proof follows the lines of the proof of Lemma 3.8: we set a = z−n andb = −e−2πiθz−m. This time the relation is a+ b+ab = 0. The proof follows by applyingthis equation to the second to last equality in (3.5.6). �

For each z ∈ C we define a map σz : C → C: if |z| < 1 we set σz(w) = w for allw ∈ C, if |z| ≥ 1 we set σz(w) = w for all w ∈ C.

Lemma 3.10. The map θ 7→ fnm(θ) is continuous on [0, 1] and differentiable on[0, 1]\Enm. If θ ∈ [0, 1]\Enm and m > 0 then

(3.5.7) f ′nm(θ) = −πi

n

∑z, Pnm(z,θ)=0

σz

(µ− 1

1 + z−n

)−1

.

Proof. The continuity of f was already mentioned at the beginning of section 4.Now say θ ∈ [0, 1]\E. By Lemma 3.6 Q(e2πiu, θ) > 0 for all u ∈ R. By compactness ofS1 we have Q(e2πiu, θ′) > 0 for all θ′ in some neighborhood of θ and all u ∈ R. Hencef is differentiable at θ by (3.4.2).

Now let us assume m > 0. By the discussion in the last paragraph we may exchangeintegration and differentiation to obtain

df

dθ=

12

d

dθ

∫ 1

0log Q(e2πiu, θ)du =

12

∫ 1

0

∂

∂θlog Q(e2πiu, θ)du

=12

∫ 1

0T (e2πiu, θ)du.(3.5.8)

We can rewrite (3.5.8) as a complex integral

(3.5.9)df

dθ=

14πi

∫|z|=1

T (z, θ)z

dz,

the path being understood as counterclockwise. The integrand has no pole on |z| = 1because θ /∈ E. This meromorphic function also has no pole at z = 0 by (3.5.3), (3.5.4),and since m ≥ 1.

As already stated in the proof of Lemma 3.8, the non-zero poles of T (X, θ)/X areprecisely the zeros of Q(X, θ). We apply the Residue Theorem to (3.5.9) to get

(3.5.10)df

dθ=

12

∑Q(z,θ)=00<|z|<1

resX=zT (X, θ)

X.

Let us now assume Q(z, θ) = 0 and 0 < |z| < 1. By (3.4.1) either P (z, θ) = 0or P (z−1,−θ) = 0. These two cases are mutually exclusive since Q has only simplezeros by Lemma 3.7. We note that in the second case P (z−1, θ) = 0 and |z−1| > 1. Todetermine the residue at z we apply Lemma 3.8 or 3.9 depending on whether P (z, θ) = 0or P (z−1, θ) = 0. Hence each term in (3.5.10) corresponds to exactly one term in thesum (3.5.7).

On the other hand, if z 6= 0 is a zero of P (X, θ), then Q(z, θ) and Q(z−1, θ) bothvanish. Since θ /∈ E one and only one of the values |z|, |z−1| is strictly less than 1. As


before any term in (3.5.7) corresponds to exactly one term in (3.5.10) by Lemma 3.8 or3.9. �

Lemma 3.11. If z ∈ C∗, θ ∈ R, and ηn = 1 with Pnm(z, θ) = Pnm(ηz, θ), thenη = 1.

Proof. Say ξ = e−2πiθ, then zn − ξzm + 1 = zn − ξηmzm + 1. So ηm = 1 and nown(k − l)−mk = 1 implies η = 1. �

Unfortunately, the absolute value of a term in the sum (3.5.7) cannot in general bebounded from above independently of n. Nevertheless we prove a vanishing result whichwill enable us to get such a bound for |f ′nm(θ)|:

Lemma 3.12. Let θ ∈ [0, 1]\Enm and m > 0, then

(3.5.11)∑

z, Pnm(z,θ)=0

(µ− 1

1 + z−n

)−1

= 0.

Proof. By Lemma 3.1 the polynomial P (X, θ) has exactly n distinct zeros for fixedθ. As z runs over these zeros, the values zn are pairwise distinct by Lemma 3.11 andthus so are

(3.5.12)(

µ− 11 + z−n

)−1

.

For fixed ξ = e−2πiθ we consider the non-zero polynomial

U = Xn − ξn(µX − 1)m((1− µ)X + 1)n−m ∈ C[X]

of degree at most n. If P (z, θ) = 0, then a calculation using zn + 1 = ξzm shows that(3.5.12) is a zero of U . Since there are n distinct values (3.5.12), these must be preciselythe zeros of U . In particular U has degree n. It follows that the sum in the assertion isa multiple of the coefficient of Xn−1 in U . But this coefficient is

−ξn(µm(1− µ)n−m−1(n−m)−mµm−1(1− µ)n−m) = 0

since µ = m/n. �

Lemma 3.13. Let θ ∈ [0, 1]\Enm, then |f ′nm(θ)| < 4π/√

3.

Proof. We eliminate the case m = 0 first. Indeed then automatically n = 1 andP = X − e−2πiθ + 1. Thus f(θ) = log max{1, |e−2πiθ − 1|} for all θ. The function f isdifferentiable on [0, 1]\{1/6, 5/6}. Using calculus one can show that the derivative of fis at most

√3π for 1/6 < θ < 5/6.

Let us move on to the case m > 0. We add πi/n times the expression on the left of(3.5.11) to f ′(θ) as in Lemma 3.10 to deduce

(3.5.13) f ′(θ) =πi

n

∑z, P (z,θ)=0

|z|>1

((µ− 1

1 + z−n

)−1

−(

µ− 11 + z−n

)−1)

.


We continue by bounding the absolute value of (3.5.13) term-wise.Let P (z, θ) = 0, |z| > 1, and let us define δ = µ− 1

1+z−n . We have zn = µ−δ1−µ+δ and

zn + 1 = 11−µ+δ = ξzm where ξ = e−2πiθ. So |1− µ + δ| < 1 and

(3.5.14) |µ− δ|m = |1− µ + δ|m−n,

thus |µ− δ| > 1. We apply the parallelogram equality |x + y|2 + |x− y|2 = 2|x|2 + 2|y|2(x, y ∈ C) with x = 1/2 and y = 1/2− µ + δ to obtain

(3.5.15) |1− µ + δ|2 + |µ− δ|2 =12

+ 2∣∣∣∣12 − µ + δ

∣∣∣∣2 = 1 + 2|µ− δ|2 + 2(Reδ)− 2µ.

We define w0 = |µ − δ|2 > 1. From (3.5.14) we deduce |1 − µ + δ|2 = w−µ/(1−µ)0 . We

insert this equality into (3.5.15) to obtain

(3.5.16) 2Reδ = w−µ/(1−µ)0 − w0 + 2µ− 1.

On the other hand w0 = |δ|2−2µReδ+µ2, hence together with (3.5.16) we get k(w0) = 0where

k(w) = µw−µ/(1−µ) + (1− µ)w − |δ|2 + µ2 − µ, w ∈ (0,∞).

Elementary calculus shows that k has a unique minimum at(µ

1− µ

)2(1−µ)

≤ 1.

Hence k is strictly increasing on [1,∞). Since w0 > 1 we deduce

0 = k(w0) > k(1) = µ2 − µ + 1− |δ|2,

and so |δ|2 > µ2 − µ + 1. But the right-hand side of this inequality is ≥ 3/4 sinceµ ∈ (0, 1/2]. We conclude

(3.5.17) |δ| >√

3/2.

Back to our derivative. Each term in (3.5.13) is of the form δ−1 − δ−1 with δ as inthe last paragraph. Thus we apply the triangle inequality and (3.5.17) to (3.5.13) tocomplete the proof. �

Finally we have:

Lemma 3.14. The map θ 7→ fnm(θ/n) is of bounded variation on [0, 1) with totalvariation bounded by 4π√

3n.

Proof. If S is a real interval [a, b] with a finite set of points removed, and F is acontinuous real-valued function on [a, b] and differentiable on S with |F ′| ≤ M , then itis well-known that the total variation of F is at most M(b − a). So the result followsfrom Lemma 3.13 with F (θ) = f(θ/n) and M = 4π√

3n. �


6. Bounding the discrepancy

Recall that φ is Euler’s totient function. Let d be a positive integer and let {xk}denote the sequence of numbers a/d where a runs over the φ(d) integers in [1, d] that arecoprime to d. We set D(d) to be the discrepancy of this sequence as defined in [Har98].That is

(3.6.1) D(d) = supI⊂[0,1)

∣∣∣∣∣∣∣(∑

1≤a≤d, (a,d)=1a/d∈I

1)− φ(d)|I|

∣∣∣∣∣∣∣ .Above I runs over all open, closed, and half-open intervals of length |I|. Let hd =1 + 1

2 + · · ·+ 1d be the dth harmonic number and let ω(d) denote the number of distinct

prime factors of d. We use a Theorem of Erdos and Turan to show:

Lemma 3.15. There exists an absolute constant c such that

(3.6.2) D(d) ≤ chdφ(d)

d

∏p|d

prime

(1 +p

p− 1) ≤ c2ω(d)hd for all d ≥ 1.

The choice c = 3 + 2/π is possible.

Proof. The second inequality in (3.6.2) follows from

φ(d)∏p|d

prime

(1 +p

p− 1) = d

∏p|d

prime

(1 +p− 1

p) ≤ d2ω(d).

We proceed to prove the first inequality in (3.6.2). The Erdos-Turan Theorem(Theorem 5.5 in [Har98] page 129) applied with L = d and N = φ(d) gives the upperbound

(3.6.3)D(d)φ(d)

≤ 1d + 1

+ 2(1 + 1/π)d∑

m=1

1mφ(d)

∣∣∣∣∣∣φ(d)∑k=1

exp(2πimxk)

∣∣∣∣∣∣ .We start by handling the Ramanujan sum

s(m) =φ(d)∑k=1

exp(2πimxk) =∑

1≤a≤d, (a,d)=1

exp(2πima/d).

We use Theorem 272 of [HW05] to evaluate

s(m) =µ(d/(d, m))φ(d/(d,m))

φ(d)


where µ is the Mobius function. We rearrange the second term on the right of (3.6.3)and obtain

d∑m=1

1mφ(d)

∣∣∣∣∣∣φ(d)∑k=1

exp(2πimxk)

∣∣∣∣∣∣ =d∑

m=1

|µ(d/(d, m))|mφ(d/(d,m))

=∑e|d

∑1≤m≤d(d,m)=e

|µ(d/e)|mφ(d/e)

=∑e|d

|µ(d/e)|eφ(d/e)

∑1≤m≤d

(d,m)=e, m′=m/e

1m′

≤ hd

∑e|d

|µ(d/e)|eφ(d/e)

.

The arithmetic function g(a) =∑

e|a|µ(a/e)|eφ(a/e) is multiplicative. If p is a prime and f is a

positive integer, then g(pf ) = p−f (1 + p/(p− 1)). Hence

g(a) =1a

∏p|a

(1 +p

p− 1) ≥ 1

a.

The lemma now follows easily from D(d)/φ(d) ≤ 1d+1 + 2(1 + 1/π)g(d)hd. �

It is well-known that 2ω(n) = O(nε) and n = O(φ(n)1+ε) for every ε > 0. Hence theexpression D(d)/φ(d) is O(d−1+ε) and especially D(d)/φ(d) → 0 as d →∞.

7. Proof of Theorem 3.1 and Corollary 3.3

Lemma 3.16. Let m,n be coprime integers with n ≥ 1, we set

N =[

n 0m −1/n

]and let F ⊂ R2 denote the image of [0, 1)2 under N . Then

(i) if x ∈ R2 there exists y ∈ Z2 with x− y ∈ F ,(ii) if x′, x′′ ∈ F with x′ − x′′ ∈ Z2 then x′ = x′′.

Proof. Let x = (x1, x2)t ∈ R2. Since m and n are coprime we have

{mk

nmod 1; 1 ≤ k ≤ n} = {k

nmod 1; 1 ≤ k ≤ n}.

We may choose an integer k with 1 ≤ k ≤ n such that

(3.7.1) x2 −m

n(x1 − [x1])−m +

mk

n= y2 −

t

n


for some t ∈ [0, 1) and some y2 ∈ Z. We define s = (x1 − [x1] + n − k)/n and notes ∈ [0, 1). The left-hand side of (3.7.1) equals x2 −ms. We set y1 = [x1] − n + k andy = (y1, y2)

t ∈ Z2. Part (i) now follows since x = y + N(s, t)t.We now prove part (ii). So let x′ and x′′ be as in the hypothesis, i.e. x′−x′′ = Nu ∈

Z2. We may assume u = (s, t)t with 0 ≤ t < 1 and |s| < 1. Now ns and ms − t/n areboth integers. We subtract n times the second expression from m times the first andconclude t ∈ Z. Hence t = 0 and so ms, ns are integers. There are integers a and b witham + bn = 1, it follows that s = ams + bns ∈ Z. Hence s = 0; we conclude x′ = x′′. �

The map θ 7→ fnm(θ/n) is continuous on [0, 1], so we may integrate.

Lemma 3.17. Let m,n be coprime integers with 0 6= n ≥ m ≥ 0, then∫ 1

0fnm(

θ

n)dθ = S0.

Proof. We have∫ 1

0f(

θ

n)dθ =

∫[0,1]2

log |e2πinu − e2πi(mu−θ/n) + 1|dudθ.

We may replace [0, 1]2 by [0, 1)2 in this equality, indeed the two sets differ by a set ofmeasure zero. We apply the linear change of coordinates u′ = nu, θ′ = mu − θ/n ofdeterminant −1 defined in Lemma 3.16. So

(3.7.2)∫ 1

0f(

θ

n)dθ =

∫F

log |e2πiu′ − e2πiθ′ + 1|dθ′du′

where F ⊂ R2 is as in Lemma 3.16. The integrand in (3.7.2) is invariant under theaction of Z2 on R2. Because of Lemma 3.16 we may replace F in (3.7.2) by [0, 1)2 andeven by [0, 1]2. Therefore

(3.7.3)∫ 1

0f(

θ

n)dθ =

∫[0,1]2

log |e2πiu′ − e2πiθ′ + 1|dθ′du′.

The translation u′ 7→ u′ + 1/2 shows that (3.7.3) is equal to the logarithmic Mahlermeasure of the two variable polynomial X + Y − 1 which is by definition just S0. �

Proposition 3.1. Let x ∈ M with D = [Q(x) : Q]. There exists a positive integerd with φ(d) ≤ D such that

|h(x, 1− x)− S0| ≤ c(d)D−1

where

c(d) =

{260 if d|6,

(4π√

3 + 8√3)2ω(d)hd otherwise.

Proof. Let α, ζ, m, n be as in Lemma 3.2, and define d = ord ζ. The proposition iseasy if h(x, 1− x) = 0. Indeed in this case x is a root of unity by Kronecker’s Theorem.By the discussion in the beginning of section 2, x is a 6th root of unity and we can taked = 1.


So we assume h(x, 1 − x) > 0. We set K = Q(ζ) and note φ(d) ≤ D. By Lemmas3.5 and 3.17 the absolute value |h(x, 1− x)− S0| equals

n

[K(α) : K]

∣∣∣∣∣∣∣ 1

[K : Q]

∑1≤a≤d(a,d)=1

f(a

dn)

−∫ 1

0f(

θ

n)dθ +

n− [K(α) : K]n

S0

∣∣∣∣∣∣∣ ,so

|h(x, 1− x)− S0| ≤n

[K(α) : K]

∣∣∣∣∣∣∣ 1

φ(d)

∑1≤a≤d(a,d)=1

f(a

dn)

−∫ 1

0f(

θ

n)dθ

∣∣∣∣∣∣∣(3.7.4)

+|n− [K(α) : K]|

[K(α) : K]S0.

Let V be the total variation of the map θ 7→ f(θ/n) over [0, 1). We apply Koksma’sinequality (Theorem 5.4 [Har98] page 124) to the first term in the upper bound of(3.7.4) to deduce

|h(x, 1− x)− S0| ≤ Vn

[K(α) : K]D(d)φ(d)

+|n− [K(α) : K]|

[K(α) : K]S0,

here D(d) is as defined in (3.6.1).By Lemma 3.14 we have V ≤ 4π√

3nand by Lemma 3.15 D(d) ≤ (3 + 2/π)2ω(d)hd,

therefore

|h(x, 1− x)− S0| ≤4π√

3(3 + 2/π)

2ω(d)hd

[K(α) : K]φ(d)+|n− [K(α) : K]|

[K(α) : K]S0

= (4π√

3 +8√3)2ω(d)hd

D+|n− [K(α) : K]|

Dφ(d)S0.(3.7.5)

There are two cases.Let us first assume that d|6. By Lemma 3.5 we have |[K(α) : K]−n| ≤ 2. Therefore

(3.7.5) implies

|h(x, 1− x)− S0| ≤ (4π√

3 +8√3)2ω(d)hd

D+ 2

φ(d)D

S0

Since d ∈ {1, 2, 3, 6} we easily bound

|h(x, 1− x)− S0| <260D

,

and hence the proposition holds for d | 6.Now let us assume d - 6. Then [K(α) : K] = n by Lemma 3.5. Therefore the

dangerously large term on the very right of (3.7.5) disappears. We have

|h(x, 1− x)− S0| ≤ (4π√

3 +8√3)2ω(d)hd

D,


which completes the proof. �

The proof of Theorem 3.1 is now a consequence of the previous proposition togetherwith standard inequalities concerning arithmetic functions.

Let ε > 0 and say x ∈ M with D = [Q(x) : Q]. Let d be as in Proposition 3.1.By Theorem 317 and §22.13 of [HW05] there exists c1(ε) independent of d such thatω(d) ≤ (1+ ε) log d

log log(3d) + c1(ε). It is well-known that there exists c2(ε), also independentof d, such that φ(d) ≥ c2(ε)d1−ε. We also recall the elementary inequality hd ≤ 1+log d.Hence

2ω(d)hd ≤ c3(ε)e(ε+log 2)

log φ(d)log log(3φ(d)) .

for some constant c3(ε) and all d ≥ 1. The theorem follows from the inequality above,Proposition 3.1, and φ(d) ≤ D. �

Finally we prove Corollary 3.3.Let x ∈ M with xr(1− x)s = 1 where r, s are integers and not both zero. Then an

easy local calculation gives max{h(x), h(1 − x)} = max{|r|,|s|}||(r,s)|| h(x, 1 − x). We have the

inequalities 12 ||(r, s)|| ≤ max{|r|, |s|} ≤ ||(r, s)|| which together with Theorem 3.1 imply

that any height value not in [S0/2, S0] is isolated. Furthermore, for any β ∈ [1/2, 1]we may choose two sequences of positive integers rn ≤ sn with (rn, sn) = 1, sn strictlyincreasing, and with sn/(rn + sn) → β. Let xn > 1 be a real with xrn

n (1 − xn)sn = 1.Then the xn must be pairwise different since they are not roots of unity. Becauseh(xn) ≤ log 2 we have [Q(xn) : Q] → ∞. By Theorem 3.1 and our choice of rn, sn wehave max{h(xn), h(1− xn)} = sn

rn+snh(xn, 1− xn). This sequence converges to βS0. �

8. Counting multiplicative dependent points

We show an easy consequence of Lemmas 3.2 and 3.5.

Lemma 3.18. Let D ≥ 1 and x ∈M(D), there exist coprime integers r, s with r ≥ 0and a root of unity η of order d such that xr(1− x)s = η and

||(r, s)|| ≤ D

φ(d)+ 2.

Proof. Let x ∈ M(D). Let k, l, m, n, r, s, α, and ζ be as in Lemma 3.2. SayR = Xn− ζ lXm + ζk and K = Q(ζ). We may assume r ≥ 0 by replacing ±ζ with ±ζ−1

if necessary.First say h(x, 1− x) = 0. By Kronecker’s Theorem α is a root of unity and we may

take r = 1, s = 0, and η = x.Now say h(x, 1 − x) > 0. We set xr(1 − x)s = η. Let d be the order of η, then

[K : Q] = φ(d) since η = ±ζ±1. Lemma 3.2 implies ||(r, s)|| = n. We have R(α) = 0and α is not a root of unity. Hence by Lemma 3.5 we conclude [K(α) : K] ≥ n− 2. Thelemma now follows from [K(α) : K] = [Q(x) : K] ≤ D/φ(d). �


Let r, s be coprime integers and η a root of unity. In Lemma 7 of [Mas07] Masserproved the bound

|{x ∈ Q\{0, 1}; xr(1− x)s = η}| ≤ ||(r, s)||.This bound together with Lemma 3.18 gives the following upper bound for the cardi-nality of M(D):

|M(D)| ≤∑d≥1

φ(d)≤D

∑r,s coprime, r≥0

||(r,s)||≤2+ Dφ(d)

∑η

ord η=d

||(r, s)||(3.8.1)

=∑d≥1

φ(d)≤D

φ(d)∑

r,s coprime, r≥0

||(r,s)||≤2+ Dφ(d)

||(r, s)||.

We follow [Mas07] and define S(X) =∑

1≤m≤X mφ(m) for real X ≥ 1.

Lemma 3.19. For X ≥ 1 we have

(3.8.2)∑

r,s coprime, r≥0||(r,s)||≤X

||(r, s)|| = 3S(X) + 1.

Proof. The proof follows by splitting the sum in (3.8.2) up into three sums overs ≥ 0, −r ≤ s < 0, and s < −r ≤ 0 respectively. The first sum equals S(X) + 1 andeach of the other two is S(X). �

Lemma 3.20. For X ≥ 1 we have S(X) ≤ 2π2 X3 + X2 log X + 2X2.

Proof. The proof is similar to the proof of Lemma 8 given in [Mas07] whichprovides an analogous lower bound for S(X).

Recall that µ denotes the Mobius function. For Y ≥ 1 we define T (Y ) =∑

1≤d≤Y d2.The identity φ(m) =

∑d|m µ(d)m/d implies

S(X) =∑

1≤d≤X

µ(d)dT (X

d)

after changing the order of summation. Let X ≥ 1, we use∑∞

d=1 µ(d)d−2 = 6/π2 toconclude∣∣∣∣S(X)− 2

π2X3

∣∣∣∣ =∣∣∣∣∣S(X)− X3

3

∞∑d=1

µ(d)d2

∣∣∣∣∣≤

∣∣∣∣∣∣∑

1≤d≤X

(µ(d)dT (

X

d)− X3

3d2µ(d)

)∣∣∣∣∣∣+ X3

3

∣∣∣∣∣∑d>X

µ(d)d2

∣∣∣∣∣≤

∑1≤d≤X

d

∣∣∣∣T (X

d)− X3

3d3

∣∣∣∣+ X3

3

∑d>X

1d2

.


Elementary calculations give |T (Y )− 13Y 3| ≤ Y 2 for Y ≥ 1. We use this inequality and∑

d>X d−2 ≤ 2X−1 to bound∣∣∣∣S(X)− 2π2

X3

∣∣∣∣ ≤ ∑

1≤d≤X

X2

d

+23X2.

The proof follows from the inequality∑

1≤d≤X d−1 ≤ 1 + log X. �

The proof of Theorem 3.3 is now simple. Certainly the lower bound in (3.1.3) followsfrom Theorem 2 in [Mas07]. We proceed to prove the upper bound.

Let D ≥ 3. By (3.8.1) and Lemma 3.19 we have

|M(D)| ≤∑d≥1

φ(d)≤D

φ(d)(3S(2 +D

φ(d)) + 1).

We apply Lemma 3.20 to this last inequality to obtain

(3.8.3) |M(D)| ≤∑d≥1

φ(d)≤D

(6π2

D3

φ(d)2+ c′

D2

φ(d)log D

),

for some absolute constant c′. We may take c′ = 200. Lemma 9 of [Mas07] gives∑d≥1

φ(d)≤D

1φ(d)

< 5(log D)2,

the fact that D may not be an integer is not important. We insert this bound into(3.8.3) to obtain

|M(D)| ≤ 6π2

D3

∑d≥1

φ(d)≤D

1φ(d)2

+ 5c′D2(log D)3 ≤ c0D3 + 1000D2(log D)3.

The proof follows from this last inequality. �

CHAPTER 4

Dependent solutions of x + y = α

This chapter extends some methods used in chapter 2: we give essentially optimalupper bounds for the height of multiplicatively dependent algebraic solutions of theinhomogeneous linear equation x+ y = α in two unknowns x and y where α is any non-zero algebraic number. Furthermore, we will study the case where α ≥ 2 is a rationalpower of a non-zero integer and derive a better bound for the height of the solution.We will also see that this bound is best possible in the case where α ≥ 2 is a rationalpower of 2 and even that the maximal height value is isolated if α is also assumed to bean integer. Moreover, for non-zero rational α we give a bound independent of α for thenumber of solutions of x + y = α if the unknowns are algebraic units in the union of allnumber fields that have unit group of rank 1.

The content of this chapter has been published in [Hab05].

1. Height bounds for dependent solutions of x + y = α

We recall that H(·) denotes the non-logarithmic absolute Weil height defined inchapter 1.

Let α be a non-zero algebraic number. In Theorem 4.1 we will show that the heightsof multiplicatively dependent algebraic numbers x, y satisfying

(4.1.1) x + y = α

are effectively bounded in terms of the height of α. More precisely:

Theorem 4.1. Let α be a non-zero algebraic number, and let x, y be non-zero mul-tiplicatively dependent algebraic numbers with x + y = α. Then

H(x, y) ≤ 2H(α)2,(4.1.2)

min{H(x),H(y)} ≤ 2H(α).(4.1.3)

The general strategy in the proof of Theorem 4.1 is the following: Given non-zeroalgebraic numbers x, y, α as in Theorem 4.1, there are integers r, s not both zero suchthat xrys = 1. Define the rational function f = T r(T − α)s and let d be the degreeof f . We note that f(x) = ±1. For any algebraic τ not a pole of f we will derive alower bound for H(f(τ))H−d in terms of H(α) and d. Here H will be either H(τ) orH(τ, τ − α) depending on the sign of rs. By substituting τ = x we will get an upperbound for H(x) independent of r, s, and d. Here it will be essential that the lower bound

43


has optimal dependency on d; we will achieve this through careful estimates of certainpositive local minima, together with the use of the product formula to avoid zero values.

In the special case α = 1 we get

(4.1.4) H(x, y) ≤ 2,

and recover the height bound from Theorem 2.1. Since max{H(x),H(y)} ≤ H(x, y),the inequality (4.1.4) also implies Theorem 1 of Cohen and Zannier’s article [CZ00].

The bound (4.1.4) and also the one from [CZ00] are easily seen to be sharp aftersetting x = y = 1/2. More generally, pick any φ ∈ Q with φ ≥ 0 and set x = y = 2−φ−1,α = 2−φ. Then x + y = α. Further, by using the standard properties of the heightfunction stated in the next section one obtains H(x, y) = 2φ+1 = 2H(α). So an upperbound for H(x, y) has to be at least linear in H(α) and (4.1.3) is sharp. We ask thequestions: can the bound for the height in (4.1.2) be improved? If so, is the upperbound linear in H(α)? The next theorem gives a positive answer to the first question.It is proved using simple ideas from diophantine approximation.

Theorem 4.2. Let α be a non-zero algebraic number, and let x, y be non-zero mul-tiplicatively dependent algebraic numbers with x + y = α. Then

H(x, y) ≤ 14H(α) log(3H(α)).(4.1.5)

Note that the estimate given in Theorem 4.2 is asymptotically much better forH(α) → ∞ then the one given in Theorem 4.1, even though it is worse for smallH(α) < 31.

Still the logarithm in (4.1.5) seems a bit disturbing, as it does not answer the secondquestion posed above. In fact a linear inequality like

H(x, y) ≤ CH(α)

with an absolute constant C is impossible - even if x, y, and α are restricted to Q - asthe following Theorem shows.

Theorem 4.3. For each pair of reals θ, Θ with 0 < θ < 1 and Θ > 1, there arenon-zero α ∈ Q with H(α) > Θ, and non-zero multiplicatively dependent x, y ∈ Q withx + y = α, such that

max{H(x),H(y)} > θH(α) log 3H(α)(log log 3H(α))2

.

So the bound given in Theorem 4.2 is optimal in the sense that it cannot be replacedby c(ε)H(α)(log 3H(α))1−ε for some ε > 0 (and similarly the bound given in Theorem4.3 cannot be replaced by c(ε)H(α)(log 3H(α))1+ε).

Until now we have considered any non-zero algebraic α, but if we restrict α to specialvalues, then we are able to improve (4.1.5) to a linear bound.

Theorem 4.4. Let α = nφ where n is a positive integer and φ a positive rational,and suppose that α ≥ 2. Then for all non-zero multiplicatively dependent algebraic

4. DEPENDENT SOLUTIONS OF X + Y = α 45

numbers x, y with x + y = α we have

(4.1.6) H(x, y) ≤ 2H(α)

with equality if and only if α is a rational power of 2 and x = 2α or y = 2α.

We already know that inequality (4.1.6) holds for α = 1 by Theorem 4.1. So (4.1.6)is valid for every non-zero integer α.

Once we have a sharp upper bound, say B(α) for H(x, y) as in Theorem 4.4, it is aninteresting problem to determine if this upper bound is isolated in the sense that thereexists ε(α) > 0 such that either

(4.1.7) H(x, y) = B(α) or H(x, y) ≤ B(α)− ε(α).

This kind of problem was first studied by Cohen and Zannier ([CZ00] Proposition 1)who proved isolation for α = 1 with B(α) = 2 and with max{H(x),H(y)} instead ofH(x, y). They used Bilu’s Equidistribution Theorem. In chapter 2 we proved that ifα = 1, then either H(x, y) = 2 or H(x, y) ≤ 1.915. We will now formulate an isolationresult if α = 2φ with φ ∈ N.

Theorem 4.5. Let α = 2φ where φ is a positive integer. Then for all non-zeromultiplicatively dependent algebraic numbers x, y with x + y = α we have either

H(x, y) = 2H(α) or H(x, y) ≤ 1.98H(α).

Theorem 4.5 shows not only that our ε(α) in (4.1.7) can be chosen independent ofα, but that it can even be chosen such that ε(α) →∞ if H(α) →∞.

We move on to an application. Given a subset M ⊂ Q we define SM (α) to be theset of pairs (x, y) ∈ M2 that satisfy (4.1.1) and where x, y are also required to be unitsin the ring of algebraic integers. Equations of this type are called unit equations andhave been studied extensively for example if M = K is a number field. In this case it isa well-known result that SK(α) is finite and even |SK(α)| ≤ c(K), i.e. the cardinalityis bounded independently of α. For example Evertse showed ([Eve84] Theorem 1)|SK(α)| ≤ 3 · 72[K:Q]+r where r is the number of real embeddings of K. Further boundsfor |SK(α)| have been obtained by Beukers and Schlickewei ([BS96] Theorem 1.1).

Now let K be a number field with unit group of rank 1 and α ∈ Q∗. If x and ysolve the unit equation, then they are multiplicatively dependent and so by Theorem4.1 their height is bounded effectively in terms of H(α). Define F to be the union ofall number fields with unit group of rank 1. Then the same argument just given leadsto an effective height bound for x, y with (x, y) ∈ SF (α). In both cases Dirichlet’s UnitTheorem shows that the degrees of x and y do not exceed 4. So Northcott’s Theoremimplies that SK(α) and SF (α) are finite sets. In fact we will prove a uniform result inthe spirit of Evertse.

Theorem 4.6. Let α be a non-zero rational number and K a number field with unitgroup of rank 1 and F as above. Then |SK(α)| ≤ 292 and |SF (α)| ≤ 755 · 106.

The first inequality is merely a numerical improvement of a special case of Evertse’sbound which leads to |SK(α)| ≤ 3 · 78 = 17294403. The second result is best possible


in the sense that if F is replaced by F ′, which is the union of all number fields withunit group of rank 2, then SF ′(α) is infinite. Indeed let a ∈ Z and let x be a zero of thepolynomial T (1 − T )(a − T ) − 1 ∈ Z[T ] which is easily seen to be irreducible. Then xand 1− x are units. Clearly there are infinitely many such x as a runs over all rationalintegers. This is not a contradiction to the Theorem above because for a large enoughx lies in a cubic number field with unit group of rank 2.

In section 2 we introduce notation used throughout the chapter and state the re-quired results on lower bounds for the height of values of special rational functions. Insection 3 we prove the statements made in section 2. These results are then used insection 4 to prove Theorems 1 and 2. In section 5 we prove Theorem 3 by construction.Finally in sections 6, 7, and 8 we prove Theorems 4, 5, and 6 respectively.

2. Notation and auxiliary results on rational functions

Let f = f(T ) be a rational function with algebraic coefficients and let τ ∈ Q. Weare interested in how H(f(τ)) depends on H(τ) and f (provided τ is not a pole off). Recall that the degree of f is defined as max{deg(P ),deg(Q)} for any two coprimepolynomials P , Q with f = P/Q. Classically it is known that

(4.2.1) C ≤ H(f(τ))H(τ)deg(f)

≤ C ′

with positive constants C, C ′ that are independent of τ (for example it follows easilyfrom Theorem 1.8 page 81 of [Lan83]).

As pointed out in section 1 we are particularly interested in a sharp lower bound of(4.2.1) for certain rational functions. The first function we will investigate has the formf = T r(T −α)s with α a non-zero algebraic number. Its degree is |r|+ |s| if rs ≥ 0, andmax{|r|, |s|} if rs < 0.

Proposition 4.1. Let r, s be integers and α a non-zero algebraic number and definethe rational function f = T r(T − α)s. Put

e = e(f) =

2|s| − |r| (rs < 0, |r| < |s|),|r| (rs < 0, |r| ≥ |s|),|r|+ |s| (rs ≥ 0)

and

H ={

H(τ) rs < 0,H(τ, τ − α) rs ≥ 0.

Then for all algebraic τ 6= 0, α we have

(4.2.2)H(f(τ))

Hd≥ 1

2d

1H(α)e

≥ 12d

1H(α)2d

.

This exponent e = e(f) in (4.2.2) looks strange, but in fact it is best possible foreach value of r and s 6= 0.

We will also study lower bounds for certain types of polynomials.


Proposition 4.2. Let m,n be integers with n > m ≥ 0, let β be an algebraic number,and define the polynomial P = Tn + βTm. Put θ = m/n. Then for all algebraic τ wehave

(4.2.3)H(P (τ))H(τ)n

≥ 1− θ

21

H(β)1/(1−θ)≥ 1

2n

1H(β)n

.

Furthermore, if n = m > 0 and β 6= −1, then

H(P (τ))H(τ)n

≥ 12n

1H(β)n

.

If β is a root of unity, then Proposition 4.2 reduces to H(P (τ))H(τ)−n ≥ (2n)−1. InProposition 4.3 we use diophantine approximation to get a lower bound independent ofn. This improvement comes at a price: the denominator in the left-hand side of (4.2.3)has to replaced by X

log X for large X = H(τ)n. So strictly speaking we are not in thesituation of (4.2.1) anymore. Nevertheless Proposition 4.3 is essential for the proof ofTheorem 4.2.

Proposition 4.3. Let m,n be integers with n > m ≥ 0, let ζ be a root of unity, anddefine the polynomial P = Tn + ζTm. Then for all algebraic τ we have

H(P (τ))H(τ)n

max{1, log(H(τ)n)} ≥ 12e

with e = 2.71828...

Compare this lower bound with the easy upper bound

H(P (τ)) = H(τm(τn−m + ζ)) ≤ 2H(τm)H(τn−m) = 2H(τ)n.

By combining the previous inequality with the inequality from Proposition 4.3, we willdeduce the following amusing consequence for the logarithmic height h(τ) = log H(τ).

Corollary 4.1. For σ ∈ Q and φ ∈ Q with 0 ≤ φ ≤ 1 take any determination ofσφ. Then

|h(σ + σφ)− h(σ)| ≤ 1 + log 2 + log max{1, h(σ)}.

3. Proofs of Propositions 4.1, 4.2, 4.3 and Corollary

Our general strategy in estimating heights H(f(τ)) where f and τ are defined asin Proposition 4.1 is to consider each local factor separately. In the first step we willconsider the case rs ≥ 0. It will actually suffice to suppose r ≥ 0 and s ≥ 0, so thatf is a polynomial. Because some results proved below remain valid in a more generalcontext we will work with an arbitrary field K containing α and τ equipped with anabsolute value | · |.

For an absolute value | · | we define

mf (τ) =max{1, |τ |r|τ − α|s}max{1, |τ |, |τ − α|}d

(4.3.1)


with d = deg(f) = r + s. The subscript f will be omitted if it is clear from the contextwhat function is meant.

Lemma 4.1. Suppose r ≥ 0, s ≥ 0. Let | · | be an ultrametric absolute value on afield K. Then for any τ ∈ K\{0, α}

mf (τ) ≥ 1max{1, |α|}d

.

Proof. First assume |τ | ≤ 1, then |τ−α| ≤ max{1, |α|}, so m(τ) ≥ max{1, |τ |, |τ−α|}−d ≥ max{1, |α|}−d and the lemma follows in this case.

Let us now assume |τ | ≥ 1. We split up into two cases.The first case is |τ | 6= |α|. This inequality implies |τ − α| = max{|τ |, |α|} so

m(τ) =max{1, |τ |r+s, |τ |r|α|s}

max{1, |τ |, |α|}d≥ 1

max{1, |α|}d

since d = r + s and after considering the two possibilities |τ | > |α| and |τ | < |α|. Hencethe lemma holds in this case.

The second case is |τ | = |α|, then

m(τ) ≥ max{1, |α|r|τ − α|s}max{1, |α|}d

= max{|α|−d,|τ − α|s

|α|s} ≥ |α|−d =

1max{1, |α|}d

.

�

We now treat the case that our absolute value is not ultrametric but satisfies theweaker triangle inequality. We obtain a slightly worse estimate than in Lemma 4.1.

Lemma 4.2. Suppose r ≥ 0, s ≥ 0. Let | · | be an absolute value on a field K. Thenfor any τ ∈ K\{0, α}

mf (τ) ≥ 1(1 + |α|)d

≥ 12d max{1, |α|}d

.(4.3.2)

Proof. The second inequality in (4.3.2) follows from

1 + |α| ≤ 2 max{1, |α|}.

We proceed by proving the first.First say |τ | ≤ 1, then |τ − α| ≤ |τ |+ |α| ≤ 1 + |α| so m(τ) ≥ max{1, |τ − α|}−d ≥

(1 + |α|)−d and (4.3.2) follows.Now let us assume |τ | ≥ 1. Say first |τ − α| ≥ |τ |, then

m(τ) =|τ |r

|τ − α|r=∣∣∣1− α

τ

∣∣∣−r≥(1 +

∣∣∣ατ

∣∣∣)−r≥ 1

(1 + |α|)r≥ 1

(1 + |α|)d,

which implies (4.3.2). Say now |τ − α| ≤ |τ |, then

m(τ) = max{1, |τ |r|τ − α|s|}|τ |−d.


The lemma follows easily if s = 0, so let us assume s > 0. If |τ | ≤ 1 + |α|; thenm(τ) ≥ |τ |−d ≥ (1 + |α|)−d by the equality for m(τ) above. If on the other hand|τ | ≥ 1 + |α|, then |τ − α| ≥ |τ | − |α| ≥ 1 and so we have

m(τ)1/s =∣∣∣∣τ − α

τ

∣∣∣∣ = ∣∣∣1− α

τ

∣∣∣ ≥ 1−∣∣∣ατ

∣∣∣ ≥ 1− |α|1 + |α|

=1

1 + |α|,

hence m(τ) ≥ (1 + |α|)−s ≥ (1 + |α|)−d, which concludes the proof. �

We will now cover the non-polynomial case of Proposition 4.1, that is if rs < 0.Recalling the definition of mf (τ) in (4.3.1) with K = C and with | · | the standardabsolute value. Taking for example r = 2 and s = −1 we obtain

mf (τ) ≤ |τ |2/|τ − α||τ |2

= |τ − α|−1,

if |τ | is large. Therefore unfortunately limτ→∞ mf (τ) = 0, so m cannot be boundedaway from 0 as in Lemma 4.2.

So we redefine mf in this case. Our motivation comes from the calculation

H(f(τ))[K:Q] =∏

v∈MK

|τ − α|−sdvv

∏v∈MK

max{1, |τ |rv|τ − α|sv}dv

=∏

v∈MK

max{|τ |rv, |τ − α|−sv }dv(4.3.3)

where we have applied the product formula (1.1.1) for the number field K if τ 6= α. Thedv are the local indices defined in chapter 1.

Now if we assume r ≥ 0 and set t = −s ≥ 0, then the new definition

(4.3.4) mf (τ) =max{|τ |r, |τ − α|t}

max{1, |τ |}dwith d = deg(f) = max{r, t}

will do the trick. We note that this time there is no extra |τ −α| in the denominator of(4.3.4).

Lemma 4.3. Suppose r > 0 > s = −t and let ε = 1− r/t. Let | · | be an ultrametricabsolute value on a field K. Then for any τ ∈ K\{0, α}

mf (τ) ≥{

max{1, |α|}−εd : if r < t and |α| = |τ | > 1max{1, |α|−1}−d : otherwise.

Proof. As in the proof of Lemma 4.1 the proof is just a study of different cases.Assume |τ | 6= |α|. Then one has |τ − α| = max{|τ |, |α|} and hence

m(τ) =max{|τ |r, |τ |t, |α|t}

max{1, |τ |d}.

If |τ | ≥ 1 then clearly m(τ) ≥ 1 so let |τ | < 1. Then

m(τ) ≥ |α|t ≥ 1max{1, |α|−1}t

≥ 1max{1, |α|−1}d


as desired.Now assume |τ | = |α|. Here

m(τ) ≥ |α|r

max{1, |α|d}.

If |α| ≤ 1 then

m(τ) ≥ |α|r ≥ 1max{1, |α|−1}d

as above.Finally if |α| > 1 then m(τ) ≥ |α|r−d. The assertion is obvious if r = d, so assume

r < d = t; then one has

m(τ) ≥ |α|−εd =1

max{1, |α|}εd.

�

Lemma 4.4. Suppose r > 0 > s = −t and let ε = 1 − r/t. Let | · | be an absolutevalue on a field K. Then for any τ ∈ K\{0, α}

mf (τ) ≥{

2−d max{1, |α|−1}−d max{1, |α|}−εd : if r < t2−d max{1, |α|−1}−d : otherwise.

Proof. We will show the slightly stronger statement

(4.3.5) m(τ) ≥{

2−d max{1, |α|}−εd : if r < t, 12 ≤

∣∣ τα

∣∣ ≤ 2, |α| ≥ 1,2−d max{1, |α|−1}−d : otherwise.

First assume that | τα | <12 or | τα | > 2. Then 1

2 max{|τ |, |α|} ≤ |τ − α|, and therefore

m(τ) ≥ 2−t max{|τ |r, |τ |t, |α|t}max{1, |τ |d}

.

If |τ | ≥ 1 then m(τ) ≥ 2−t ≥ 2−d and we are done. Assume that |τ | < 1. Then

m(τ) ≥ 2−t|α|t ≥ 2−d max{1, |α|−1}−d

which implies (4.3.5).Now assume that 1

2 ≤∣∣ τα

∣∣ ≤ 2. If |τ | ≤ |α| then

m(τ) ≥ |τ |r

max{1, |α|d}≥ 2−r |α|r

max{1, |α|d},

and if |τ | > |α| then

m(τ) ≥ 2−d |α|r

max{1, |α|d};

so in both cases we have m(τ) ≥ 2−d|α|r max{1, |α|}−d. If |α| < 1 then

m(τ) ≥ 2−d|α|r ≥ 2−d max{1, |α|−1}−d


as above. Suppose |α| ≥ 1. Then m(τ) ≥ 2−d|α|r−d. If r = d our assertion is obvious.Finally if r < d then t = d and we have

m(τ) ≥ 2−d|α|−εd = 2−d max{1, |α|}−εd.

�

Proof of Proposition 4.1: One may assume r ≥ 0 and s ≥ 0 if r = 0 (apply property1.1.5 if necessary). Let K be any number field containing τ and α, and say v ∈ MK .

If rs ≥ 0, then s ≥ 0. Lemmas 4.1 and 4.2 together imply

max{1, |τ |rv|τ − α|sv}max{1, |τ |v, |τ − α|v}d

≥ 1δv(2)d max{1, |α|v}d

.

Now the Proposition in the case rs ≥ 0 is proved by raising to the dvth power, takingthe product over all elements of MK and extracting the [K : Q]th root.

If rs < 0, then s < 0. We set t = −s and also ε = 1− r/t; now Lemmas 4.3 and 4.4together imply

max{|τ |rv, |τ − α|tv}max{1, |τ |dv}

≥{

δv(2)−d max{1, |α|v}−εd max{1, |α|−1v }−d : r < t,

δv(2)−d max{1, |α|−1v }−d : r ≥ t.

Recall (4.3.3); now the Proposition in the case rs < 0 is proved by raising to the dvthpower, taking the product over all elements of MK and extracting the [K : Q]th root.�

We now turn to the proof of Proposition 4.2. It follows the same idea as the proofof Proposition 4.1; that is, each factor in the height is estimated separately. Againthe following two lemmas hold in a more general context where K is any field with anabsolute value | · |. For θ ∈ (0, 1) and β, z ∈ K define

(4.3.6) mθβ(z) =max{1, |z|θ/(1−θ)|z + β|}

max{1, |z|1/(1−θ)}The subscripts θ and β will be omitted if the context makes it clear what is meant.

As in the previous section we shall estimate the finite part of the height functionfirst.

Lemma 4.5. Let | · | be an ultrametric absolute value on a field K. Then for anyz ∈ K

mθβ(z) ≥ 1max{1, |β|}1/(1−θ)

.

Proof. The case where |z| ≤ 1 is trivial because then m(z) ≥ 1. If |z| > 1we split into three cases. First if |β| < |z| then m(z) = 1. Next if |β| = |z| thenm(z) = max{|z|−1/(1−θ), |1 + β/z|} ≥ |z|−1/(1−θ) = |β|−1/(1−θ). And finally if |β| > |z|then m(z) = |β|/|z| > 1.

�

Finally we will give an estimate of the factors in the infinite part of the heightfunction in the next lemma.


Lemma 4.6. Let | · | be an absolute value on a field K. Then for any z ∈ K

mθβ(z) ≥ 1− θ

21

max{1, |β|}1/(1−θ).

Proof. We split up the proof into two parts the first one being when

(4.3.7)1− θ

2|z|1/(1−θ) max{1, |β|}−1/(1−θ) ≤ 1.

But then one has

m(z) ≥ 1max{1, |z|}1/(1−θ)

≥ 1− θ

21

max{1, |β|}1/(1−θ)

which is just the assertion. Now assume that (4.3.7) does not hold. This is equivalentto saying |z| > φµ with

φ =(

21− θ

)1−θ

and µ = max{1, |β|}.

Obviously one has |z| > 1 and therefore

m(z) ≥ |z|θ/(1−θ)|z + β||z|1/(1−θ)

=|z + β||z|

≥ 1− |β||z|

≥ 1− µ

|z|

> 1− 1φ≥(

1− 1φ

)1

µ1/(1−θ).

The proof of the lemma is complete if the inequality

(4.3.8) 1−(

1− θ

2

)1−θ

= 1− 1φ≥ 1− θ

2

holds. Set ξ = 1− θ then (4.3.8) is equivalent to

(4.3.9) g(ξ) ≤ 1 for all ξ ∈ (0, 1) with g(ξ) =(

ξ

2

)ξ

+ξ

2.

Note that defining g(0) = 1 makes the map continuous on [0, 1] and

d2g

dξ2=(

ξ

2

)ξ ((log(ξ/2) + 1)2 +

1ξ

)> 0

for ξ ∈ (0, 1). So g is convex which implies (4.3.9) because g(0) = g(1) = 1. �

Note that Lemma 4.6 is in some cases an improvement of Lemma 4.2. Indeed recallthe definition of f and assume r ≥ 0, s > 0 so d = r+s > 0. Set θ = r/d, ξ = 1−θ = s/d.Then Lemma 4.6 with β = −α, in conjuction with (4.3.1), (4.3.6) imply

mf (z) = mθα(z)d(1−θ) ≥(

1− θ

2

)d(1−θ) 1max{1, |α|}d

and the latter is larger the (2 max{1, |α|})−d if and only if (ξ/2)ξ > 12 , that is, ξ < 1

2 .


We now prove Proposition 4.2. Let K be a number field containing β, τ and letv ∈ MK . The case m = 0 follows directly from property 1.1.6 of the height function soone may assume θ = m/n > 0. First consider n > m; then

max{1, |τn + βτm|v}max{1, |τ |nv}

=max{1, |z|θ/(1−θ)

v |z + β|v}max{1, |z|1/(1−θ)

v }for z = τn−m. We apply Lemmas 4.5 and 4.6, raise to the dvth power, take the productover all elements of MK and then the [K : Q]th root to prove the first inequality of(4.2.3). The second inequality follows immediately from the fact that 1/(1− θ) ≤ n.

Finally if n = m and β 6= −1 then standard height properties implyH(P (τ))H(τ)n

≥ 1H(1 + β)

≥ 12H(β)

.

�

To prove Proposition 4.3 we will apply diophantine approximation to Proposition4.2. This idea came up in a private correspondence Bombieri-Masser-Zannier June 2002.

Set θ = m/n. The case m = 0 can be dismissed as trivial so we may assumeθ ∈ (0, 1). We would like to apply diophantine approximation to θ to create a newpolynomial with small degree.

Recall that for any Q > 1 there exists a pair p, q ∈ Z such that

|θq − p| ≤ 1Q

and 0 < q < Q.

For a reference see [Cas57] page 1. Because θ ∈ (0, 1) one has

θq − p ≤ 1Q

so p ≥ θq − 1Q

> θq − 1 > −1

thus p ≥ 0; and furthermore

p− qθ ≤ 1Q

so p ≤ 1Q

+ qθ < 1 + qθ < 1 + q

which implies p ≤ q.We choose any u ∈ Q∗ with uq = τn. We set k = mq − np and choose a β ∈ Q∗

with βq = ζτk. Note that (βup)q = βqupq = ζτk+np = ζτmq, so τm = ηβup for someroot of unity η. Now

P (τ) = τn + ζτm = uq + ξβup

for a root of unity ξ = ζη. If p = q and ξβ = −1 then P (τ) = 0 so τn−m = −ζ so τ is aroot of unity and the required result follows trivially. If p < q or ξβ 6= −1 we can applyProposition 4.2 to get

H(P (τ))H(τ)n

=H(uq + ξβup)

H(u)q≥ 1

2qH(ξβ)q≥ 1

2QH(β)q

=1

2QH(τ)|k|≥ 1

2QH(τ)n/Q


because |k| = n|θq − p|. Therefore one has

(4.3.10) H(P (τ)) ≥ H(τn)1−1/Q

2Q

for each Q > 1. Because the lower bound is continuous Q = 1 is also allowable.Optimization of the right hand side leads to the choice

Q0 = max{1, log H(τn)} ≥ 1.

Inserting Q0 into (4.3.10) gives the bound

H(P (τ)) ≥{

H(τn)(2e log H(τn))−1 : if H(τn) > e1/2 : otherwise

≥ H(τn)2emax{1, log H(τn)}

as desired. �

To prove the Corollary we note that the bound h(σ +σφ)−h(σ) ≤ log 2 has alreadybeen shown in section 2. For the corresponding lower bound we note that the case φ = 1is covered by elementary height properties; so assume φ < 1. Choose m, n ∈ Z withφ = m/n; then there is τ ∈ Q∗ and ζ a root of unity with σ = τn and σφ = ζτm. Weapply Proposition 4.3 to get

h(σ + σφ)− h(σ) ≥ −1− log 2− log max{1, h(σ)},thus concluding the proof. �

4. Proofs of Theorems 4.1 and 4.2

Before we prove Theorems 4.1 and 4.2 we need a simple lemma.

Lemma 4.7. Say x, y ∈ Q∗ with H(x) ≥ H(y) and xrys = 1 for integers r and ssuch that rs < 0, then H(x, y) = H(x). Furthermore, if H(y) > 1, then |s| ≥ |r|.

Proof. Let K be a number field containing x and y. If H(y) = 1, then |y|v = 1 forall v ∈ MK and the lemma follows. From now on we assume H(y) > 1. By invertingxrys = 1 we may also assume r > 0. Therefore s < 0 and −r/s > 0. By functionalproperties of the height function we have H(x) = H(y)−s/r ≥ H(y) and so −s ≥ rbecause H(y) > 1. The second assertion of the lemma follows. To prove the first we notethat for any v ∈ MK we have max{1, |x|v, |y|v} = max{1, |x|v, |x|−r/s

v } = max{1, |x|v}since −r/s ∈ (0, 1]. These local equalities imply H(x, y) = H(x). �

The proof of Theorem 4.1 is a simple task with the help of Proposition 4.1. Letfor example xrys = 1 with r, s ∈ Z not both zero and say H(x) ≥ H(y). We applyProposition 4.1 to ±1 = xr(x − α)s using H(±1) = 1. If rs ≥ 0 we have H(y) ≤H(x, y) ≤ 2H(α), and the theorem follows. So say rs < 0, then H(x, y) = H(x) byLemma 4.7 and furthermore H(x) ≤ 2H(α)2 by Proposition 4.1. It remains to showthat H(y) ≤ 2H(α), and for this we may clearly assume H(y) > 1. By Lemma 4.7 we


conclude |s| ≥ |r|. Finally we apply Proposition 4.1 to ys(y − α)r = ±1 and concludeH(y) ≤ 2H(α). �

We now prove Theorem 4.2. Let for example xrys = 1 with r, s ∈ Z not both zeroand assume r ≥ 0 and H(x) ≥ H(y). If rs ≥ 0 the same argument as in the proofof Theorem 4.1 gives H(x, y) ≤ 2H(α), and of course this estimate is better than ourassertion. Hence let us assume rs < 0, which implies H(x, y) = H(x). Hence it sufficesto show (4.1.5) with H(x, y) replaced by H(x).

If H(y) = 1, then H(x) = H(α− y) ≤ 2H(α) by elementary height inequalities. Sosay H(y) > 1, then Lemma 4.7 implies |s| ≥ |r|. We may even assume |s| > |r|, for ifequality were to hold, then H(x) = H(y) ≤ 2H(α) by Theorem 4.1. We set n = |s| andm = r; then there exists τ ∈ Q∗ with x = τn and y = ζτm and for some root of unityζ. Hence we have P (τ) = α with P = Tn + ζTm. Now Proposition 4.3 implies

(4.4.1) H(α) ≥ ϕ(H(x)) with ϕ(z) =z

2emax{1, log z},

ϕ being understood as a continuous map on [1,∞). Note that ϕ is increasing which canbe easily verified by restricting it to [1, e] and [e,∞). Theorem 4.2 follows if

(4.4.2) ϕ(z0) ≥ H(α) with z0 = 14H(α) log(3H(α))

holds. Indeed (4.4.1) combined with (4.4.2) leads to ϕ(z0) ≥ ϕ(H(x)) and further toz0 ≥ H(x) because ϕ is increasing.

Because z0 > e the inequality (4.4.2) is equivalent to

(4.4.3)7e

log(3w)− log(14w)− log log(3w) ≥ 0 with w = H(α)

which certainly holds for w = 1. The derivative of the left-hand side of (4.4.3) withrespect to w is

1w

(7e− 1− 1

log(3w)

)≥ 1

w

(7e− 2)

if w ≥ 1. The right-hand side is positive for every w ≥ 1 so we may conclude that(4.4.3) holds for every α thus completing the proof. �

5. Dependent solutions with large height

Choose a large integer q, and define

x =(

q

q − 1

)n

, y = −(

q

q − 1

)n−1

with n = [q log q],

so that H(x) = qn > qn−1 = H(y). Then x + y = α with α = qn−1/(q − 1)n. NowH(α) = max{qn−1, (q − 1)n} so

(4.5.1) limq→∞

log H(α)n log q

= limq→∞

max{

n− 1n

,log(q − 1)

log q

}= 1


and one clearly has

(4.5.2) limq→∞

n log q

q(log q)2= 1.

We multiply (4.5.1) and (4.5.2) to get

log H(α) = q(log q)2κ(q) with limq→∞

κ(q) = 1.

Taking the logarithm gives log log H(α) = log q + 2 log log q + log κ(q) which leads us to

(4.5.3) limq→∞

log log H(α)log q

= 1.

We want to show that the quotient H(x)/H(α) is large, so we will evaluate the limit of

(4.5.4) q−1 H(x)H(α)

= min{1, q−1(1− q−1)−n}

for q →∞. Now

log(q−1(1− q−1)−n)

= − log q − n log(1− q−1

)= − log q + n

(q−1 + O(q−2)

)= − log q + (q log q + O(1))

(q−1 + O(q−2)

)= O

(log q

q

)for q →∞. By inserting this expression into (4.5.4) one obtains

(4.5.5) limq→∞

q−1 H(x)H(α)

= 1.

Finally, by combining (4.5.1), (4.5.2), (4.5.3), and (4.5.5) we conclude

limq→∞

H(x)H(α) log H(α)(log log H(α))2

= 1.

�


The next lemma will take care of some special cases of Theorem 4.4.

Lemma 4.8. Let α = nφ ≥ 2 with n ∈ N, φ a positive rational and x, y ∈ Q∗ andx + y = α.(i) If r, s ∈ Z are not both zero with rs ≥ 0 and xrys = 1, then

H(x, y) ≤ 32H(α).

(ii) If y = ζx for some root of unity ζ, then

H(x, y) ≤√

2√

3H(α).


Proof. We can and will assume H(x) ≥ H(y). Let K be a number field containingx and y. For part (i) we assume r ≥ 0, and if r = 0 also s > 0. Lemmas 4.1 and 4.2applied to f = T r(T − α)s and τ = x give∏

v∈MK

1max{1, |x|v, |x− α|}dvd

≥∏v|∞

1(1 + |α|v)dvd

∏v-∞

1max{1, |α|v}dvd

with d = r + s. For a finite place v we have |α|v ≤ 1 and therefore H(x, y) ≤ 1 + α ≤32α ≤ 3

2H(α).Now to part (ii): We note that H(x, y) = H(x) by Lemma 4.7. We have x(1+ζ) = α

and by hypothesis ζ 6= −1; elementary height properties lead to H(x) = H((1+ζ)−1α) ≤H((1 + ζ)−1)H(α) = H(1 + ζ)H(α) ≤

√2√

3H(α) by Lemma 2.2 in chapter 2 if ζ 6= 1.So now assume ζ = 1, then x = y = α/2 and so

H(x, y)[K:Q] = H(x)[K:Q] =∏

v∈MK

max{1, |α/2|dvv }

=∏v|∞

|α/2|dvv

∏v-∞

max{1, |α/2|dvv } ≤ (α/2)[K:Q]

∏v-∞

|2|−dvv ,

which implies H(x, y) ≤ H(α). �

Lemma 4.9. Let α = nφ ≥ 2 with n ∈ N, φ a positive rational and let x, y ∈ Q∗

with x+y = α and xr = yt where 0 < r < t are rational integers. Define λ = t/r and letK be a number field containing x, y. Then x is an algebraic integer and furthermore:(i) If v is a finite place of K with |x|v < 1, then |x|v = |y|λv = |α|λv .(ii) One has

(4.6.1) H(x) =∏v-∞|x|v<1

max{1, |α|−1v }dvλ/[K:Q] ≤ αλ.

(iii) Let ε ∈ [0, 1) and δ ∈ (1, 2] be such that

(4.6.2) λ ≥ 1 +log 2log α

(1 + ε) and (δ − 1)(1− δ−1)log α

(1+ε) log 2 α ≥ 1.

Then

(4.6.3) |x|v ≤ δ|α|v for all v | ∞ and H(x) ≤ δH(α).

Proof. Note that x is an algebraic integer. Indeed α is an algebraic integer and xis a zero of the monic polynomial (−1)t(α− T )t − (−1)tT r ∈ OK [T ].

Let v be as in part (i); then xr = yt implies |y|v > |x|v and so because of x + y = α

we have |α|v = |y|v = |x|r/tv < 1.

For part (ii) recall the definition (4.3.4) of m = mf with f defined as in Proposition4.1 with s = −t. For any finite place v ∈ MK one has

(4.6.4) m(x) = max{|x|rv, |x− α|tv} = |x|rv ≤ 1.


If on the other hand v is an infinite place one has |α|v = |n|φv = α ≥ 2 so clearly |x|v ≥ 1and therefore

(4.6.5) m(x) = |x|r−tv .

Using (4.6.4), (4.6.5) and applying the product formula, we conclude

(4.6.6)∏

v∈MK

m(x)dv =∏v-∞

|x|dvrv

∏v|∞

|x|dv(r−t)v =

∏v-∞

|x|dvtv =

∏v-∞|x|v<1

|x|dvtv .

Then by applying part (i) to (4.6.6) and using (4.3.3) with f(x) = ±1 we get1

H(x)[K:Q]t=∏

v∈MK

m(x)dv =∏v-∞|x|v<1

|α|dvλtv =

∏v-∞|x|v<1

max{1, |α|−1v }−dvλt

≥ H(α−1)−[K:Q]λt,

from which (ii) follows at once.To prove part (iii) assume (4.6.2) holds. Note that the statement on the left-hand

side of (4.6.3) implies the inequality on the right-hand side because x is an algebraicinteger. We will prove the left-hand side of (4.6.3) by contradiction. Assume |x|v > δ|α|vfor some v | ∞. Then |x−α|v > |x|v(1− δ−1), and so |x|rv = |x−α|tv > (|x|v(1− δ−1))t.The first inequality in (4.6.2) implies

δα < |x|v < (1− δ−1)−λ

λ−1 ≤ (1− δ−1)−(1+ log α(1+ε) log 2

)

which contradicts the second inequality in (4.6.2). �

We can now prove Theorem 4.4. Let for example xrys = 1 with r, s ∈ Z not bothzero and assume r ≥ 0,H(x) ≥ H(y). Fix a number field K with x, y ∈ K. We willbegin by proving the inequality (4.1.6). If rs ≥ 0 or r = −s or y is a root of unity,then Lemma 4.8 implies (4.1.6). So assume rs < 0 and −s 6= r and y not a root ofunity. Then H(x, y) = H(x) and −s > r by Lemma 4.7. Hence it suffices to proveH(x) ≤ 2H(α).

For brevity we will set t = −s and λ = t/r. If λ ≤ 1 + log 2/ log α then (4.6.1) inLemma 4.9 gives

(4.6.7) H(x) ≤ αλ ≤ α1+log 2/ log α = 2H(α).

On the other hand if λ > 1 + log 2/ log α, there exists an ε > 0 such that the firstinequality in (4.6.2) holds. Now the second inequality in (4.6.2) holds strictly whenδ = 2, and so it must continue to hold for some δ < 2. Hence the left-hand side of(4.6.3) holds with some δ < 2 and therefore H(x) < 2H(α)

Finally we prove that H(x, y) = 2H(α) holds if and only if α is a rational powerof 2 and x = 2α or y = 2α. The “if” part is trivial. For the “only if” part let usassume H(x, y) = 2H(α) and H(x) ≥ H(y). As above we use Lemma 4.8 to reduceto the case rs < 0 and t = −s > r. Let again λ = t/r. By Lemma 4.7 we haveH(x, y) = H(x) = 2H(α). We have already showed that λ > 1 + log 2/ log α implies


H(x) < 2H(α). But if λ < 1 + log 2/ log α then (4.6.7) also implies H(x) < 2H(α). Sowe must have

(4.6.8) λ = 1 + log 2/ log α

and thus α is a rational power of 2. Because of (4.6.8) the choice ε = 0, δ = 2 satisfiesthe hypothesis of Lemma 4.9(iii). We conclude |x|v ≤ 2|α|v for infinite places v. As α,x are algebraic integers we even have |x|v = |2α|v for all infinite places v. Note that(4.6.8) implies

(4.6.9) αλ = 2α.

If v is a finite place with |2|v < 1 then |x|v < 1; indeed we must have equality in (4.6.1).So Lemma 4.9(i) gives |x|v = |α|λv , and because of (4.6.9) we conclude |x|v = |2α|v. Butthis last equation holds for any finite place: indeed if |2|v = 1 then |x|v = |α|v = 1because of Lemma 4.9(i). Hence |x|v = |2α|v for all places finite or infinite: thereforex = 2αξ for a root of unity ξ. Let v be a infinite place. Then the equality |x|rv = |x−α|tvand (4.6.9) imply |2ξ−1|v = 1. For any z, w ∈ K we have the equality |z+w|2v+|z−w|2v =2|z|2v + 2|w|2v; We take z = ξ − 1 and w = ξ to conclude ξ = 1. Thus x = 2α. �


We will start with a lemma concerning an elementary estimate.

Lemma 4.10. Let φ ∈ N and 1 + 23φ−1 ≤ λ ≤ 1 + 4

3φ−1, then

(4.7.1)12

max{1, λ(1 + φ−1)−1} log(22φ+1 + 2 max{1, (λ/2)1

1−λ }) ≤ log(1.98 · 2φ).

Proof. Let g(λ) denote the left-hand side of (4.7.1). The map λ → (λ/2)1/(1−λ)

decreases for 1 < λ < 4. If φ = 1 the lemma follows easily by considering the casesλ ≤ 2 and λ > 2. We will therefore assume φ ≥ 2. Since (1 + 1/w)w increases for w ≥ 1we conclude

(λ/2)1

1−λ ≤ 23φ/2

(1 + 23φ)3φ/2

≤ 33

26· 23φ/2.

If x and y are positive then log(x + y) ≤ xy + log y, thus

g(λ) ≤ 12

max{1,λ

1 + φ−1} log(

33

2623φ/2+1 + 22φ+1)

≤ max{1,λ

1 + φ−1}(3

3

28+

12

log 22φ+1).

If λ ≤ 1 + φ−1, then

g(λ) ≤ 33

28+

12

log 2 + log 2φ


and we are done. On the other hand if λ > 1 + φ−1, then

g(λ) ≤φ + 4

3

φ + 1(33

28+

12

log 22φ+1) =φ(5

6 log 2 + 33

28 ) + 32

26 + 23 log 2

φ + 1+ log 2φ

<56

log 2 +33

28+ log 2φ.

�

We proceed as follows: let x and y = α − x be multiplicative dependent withH(y) ≤ H(x) < 2H(α), and let P ∈ Z[T ] be the minimal polynomial of x. Then wewill show P (2α) 6= 0. With the help of the finite places lying above 2 we will even finda lower bound for |P (2α)| in terms of H(x). More precisely:

Lemma 4.11. Let α = 2φ for φ ∈ N and let x, y ∈ Q∗ with x + y = α and xr = yt

where 0 < r < t are rational integers. Define λ = t/r and k(w) = 2α2 + 2w2/λ − w2. IfH(x) < 2H(α), then

log H(x) ≤ 12

max{1, λ(1 + φ−1)−1} log supw≥1

k(w).

Proof. Fix a finite galois extension K/Q with galois group G such that x ∈ K.Let P ∈ Z[T ] be the minimal polynomial of x; then because x is an algebraic integer wehave

(4.7.2) P [K:Q(x)] =∏σ∈G

(T − σx).

Let v be any finite place of K extending the 2-adic absolute value i.e. v | 2, then

|P (2α)|[K:Q(x)]v =

∏σ∈G

|2α− σx|v ≤∏σ∈G

max{|2|1+φv , |σx|v}.

Recall from Lemma 4.9(i) that if v′ - ∞ then |x|v′ < 1 implies |x|v′ = |α|λv′ . Let g bethe number of finite places of K lying over 2 and g′ be the number of finite places v′

with |2|v′ , |x|v′ < 1. Now G acts transitively on the set of all finite places lying over 2(Proposition 11 page 12 of [Lan94]) therefore each stabilizer has cardinality [K : Q]/g,so

|P (2α)|[K:Q(x)]v ≤

∏σ∈G

max{|2|1+φσ−1v

, |x|σ−1v} =∏v′|2

max{|2|1+φv′ , |x|v′}[K:Q]/g

=∏v′|2

|x|v′<1

max{|2|1+φv′ , |α|λv′}[K:Q]/g = |2|

[K:Q] g′g

min{1+φ,φλ}v .

Recall that α ∈ Q, so if P (2α) = 0 then x = 2α. In this case H(x) = 2H(α), whichcontradicts our hypothesis. We conclude P (2α) 6= 0. Now P (2α) ∈ Z and the product


formula imply

|P (2α)|[K:Q] =∏v|∞

|P (2α)|dvv =

∏v-∞

|P (2α)|−dvv ≥

∏v|2

|P (2α)|−dvv(4.7.3)

≥ 2[K:Q] g′g

min{1+φ,φλ} deg(P ).

Because K/Q is galois we have dvg = [K : Q] for any v | 2. So Lemma 4.9(ii) yields

H(x) =∏v-∞

|x|v<1,|2|v<1

max{1, |α|−1v }dvλ/[K:Q] = 2

g′g

φλ.

This equation inserted into (4.7.3) gives

(4.7.4) |P (2α)|[K:Q] ≥ H(x)[K:Q]min{1,λ−1(1+φ−1)} deg(P ).

We continue by bounding the left hand side of (4.7.4) from above. Let v be an infiniteplace. Recall that for z1, z2 ∈ K we have |z1+z2|2v+|z1−z2|2v = 2|z1|2v+2|z2|2v. Let σ ∈ G,take z1 = σα, z2 = σα− σx in the previous equation and recall |σα− σx|v = |σx|1/λ

v toconclude

(4.7.5) |2α− σx|2v = |2σα− σx|2v = 2|σα|2v + 2|σx|2/λv − |σx|2v = k(|σx|v).

Note that |σx|v ≥ 1; indeed if |σx|v < 1 then |σy|v = |σx|1/λv < 1 and so |α|v < 2, a

contradiction. Apply (4.7.5) to (4.7.2) to get

|P (2α)|[K:Q(x)] =∏σ∈G

|2α− σx|v =∏σ∈G

k(|σx|v)1/2 ≤ (supw≥1

k(w))[K:Q]/2.

We combine the previous upper bound with the lower bound in (4.7.4) to conclude theproof. �

We can now prove Theorem 4.5. Assume H(x) ≥ H(y) and H(x, y) < 2H(α),furthermore let r, s ∈ Z not both zero with r ≥ 0 such that xrys = 1. With the help ofLemma 4.8 we reduce to the case t = −s > r > 0 and H(x, y) = H(x) as we did in thebeginning of the proof of Theorem 4.4. It suffices to show H(x) ≤ 1.98H(α). Defineλ = t/r; we will split up into cases.

The two first cases λ < 1+ 23φ−1 and λ > 1+ 4

3φ−1 are effectively covered in Lemma4.9. Indeed part (ii) applied to the first case gives H(x) ≤ αλ ≤ 22/3H(α). For thesecond case set ε = 1/3 and δ = 1.9. This choice clearly satisfies the left-hand side of(4.6.2). Because

(δ − 1)(1− δ−1)φ

1+ε α =910

(2(

919

)3/4

)φ

≥ 1

the right-hand side of (4.6.2) holds as well. Thus H(x) ≤ 1.9H(α).Now assume 1 + 2

3φ−1 ≤ λ ≤ 1 + 43φ−1. Let k be the function defined in Lemma

4.11. Elementary calculus yields

supw≥1

k(w) = k(w0) with w0 = max{1, (λ/2)λ

2(1−λ) }.


Apply the previous inequality to Lemma 4.11 and obtain the bound

log H(x) ≤ 12

max{1, λ(1 + φ−1)−1} log k(w0)

≤ 12

max{1, λ(1 + φ−1)−1} log(2α2 + 2 max{1, (λ/2)1

1−λ }).

Now Lemma 4.10 implies H(x) ≤ 1.98H(α). �


The next lemma proves the first inequality in Theorem 4.6.

Lemma 4.12. Let K be a number field with rankO∗K = 1 and α ∈ Q∗; then |SK(α)| ≤

292.

Proof. If α /∈ Z then SK(α) is empty so assume α ∈ Z \ {0}. Let ω be the numberof roots of unity in K, η a fundamental unit, R the regulator, and D the degree of K.

If (x, y) ∈ SK(α) there exist r, s ∈ Z not both zero such that xrys = 1; we shallfurthermore assume that r ≥ 0 and H(x) ≥ H(y) (so we will have to multiply thenumber of solutions under this hypothesis by 2 to get a bound for the total number ofsolutions). If H(y) = 1 then y is a root of unity, hence in this case we can choose r = 0,s > 0. So in all cases one may assume |s| ≥ r.

First assume α 6= ±1. Let ∆ denote the set of all infinite places v for which |x|v ≥ 1and let δ ∈ [0, 1] with δD =

∑v∈∆ dv. Note that in the case s < 0 one has δ = 1 because

|α| ≥ 2. Because x is a unit the height is given by

(4.8.1) H(x) =∏v∈∆

|x|dv/Dv .

Let v ∈ ∆; if |x|v < |α|v/2, then |x|−r/sv = |α − x|v ≥ |α|v − |x|v > |x|v which

leads to r/s < −1 so |r/s| > 1 contradicting |r| ≤ |s|. We conclude |x|v ≥ |α|v/2 =max{1, |α|v}/2 for all v ∈ ∆. So by (4.8.1)

(4.8.2) H(x) ≥ 2−δH(α)δ.

We now deduce a corresponding upper bound for H(x). If s ≥ 0, then Lemma 4.2with f = T r(T − α)s and τ = x applied to (4.8.1) leads to

(4.8.3) H(x) ≤ 2δH(α)δ.

If on the other hand s < 0, then (4.8.3) also holds because of Theorem 4.4 and δ = 1.With the bounds (4.8.2) and (4.8.3) we can apply a gap principle. There exists a

unique a ∈ Z and a root of unity ζ such that x = ζηa. We apply height functionalproperties (cf. chapter 1) and the bounds to see that |a| lies in an interval of length

δ log 4log H(η) . Hence there are at most 2(log 4/ log H(η) + 1) possibilities for a. Clearly


this estimate remains valid for α = ±1 because in this case Theorem 4.1 implies 0 ≤log H(x) ≤ log 2. We also note that R = D log H(η) and therefore

|SK(α)| ≤ 4ω

(D log 4

R+ 1)

.

Elementary considerations lead to D ≤ 4 and ω ≤ 12. Now a result of Friedman ([Fri89]Theorem B) which states R/ω ≥ 0.09058 completes the proof. �

To prove Theorem 4.6 let (x, y) ∈ SF (α). The proof splits up into two cases:(i) There exist n ∈ Z and ζ ∈ F a root of unity such that x = ζyn with−2 ≤ n ≤ 2

or y = ζxn with n = 0,±2.(ii) Otherwise.

First assume case (i). Elementary arguments show that there are 24 roots of unityζ in F . For each such ζ, substituting y = α− x in (i) gives eight polynomial equationsin x of degree at most 3; thus the number of x is at most 24 · 8 · 3 = 576.

Now assume case (i) does not hold. Set K = Q(x) = Q(x, y) and D = [K : Q].Because x and y are not roots of unity we have rankO∗

K = 1. Let η be a fundamentalunit of K. We claim that H(η)D ≤ 4 will complete the proof. Indeed assuming thisinequality a well-known argument bounds the number of units with degree d and heightat most 41/d by 2d

∏d−1k=1(2 ·

(dk

)·4+1). Take the sum over this expression for 2 ≤ d ≤ 4;

thus there are at most 430706 possibilities for η. There are 6 roots of unity in F suchthat one of these generates the group of roots of unity in K. Let ζ be such a root ofunity, then K = Q(η, ζ). Now the Theorem follows from Lemma 4.12 applied to thefield K.

We will now show H(η)D ≤ 4. There are a, b ∈ Z and roots of unity ζ, ξ such thatx = ζηa and y = ξηb. Let σ1, σ2 be two distinct non-conjugate embeddings of K intoC. These correspond to the two infinite places of K. Define di = 1 if σi(K) ⊂ R anddi = 2 otherwise. We may assume d1 ≥ d2. Now |σi(η)| 6= 1 for i = 1, 2 by Kronecker’sTheorem, so by replacing η with η−1 if necessary we may assume |σ1(η)| > 1. Let li bea logarithm of σi(η). Note that D log H(η) = d1 log |σ1(η)| = d1Re(l1), hence it sufficesto show Re(l1) ≤ log 2. The equality |σ1(η)|d1 |σ2(η)|d2 = 1 implies

(4.8.4) d1Re(l1) + d2Re(l2) = 0.

Because α ∈ Q one has σ1(x)− σ2(x) = σ2(y)− σ1(y). Apply (4.8.4) to get

(4.8.5) |eq(a)|a|Re(l1)+γ1i − e−q(−a)|a|Re(l1)+γ2i| = |eq(b)|b|Re(l1)+γ3i − e−q(−b)|b|Re(l1)+γ4i|

where the γi are real numbers and

q(w) ={

1 : w ≥ 0,d1d2

: w < 0

is an integer. Now define k = q(b)|b| − q(a)|a| ∈ Z, then k 6= 0 because otherwise wewould be in case (i). So first assume k ≥ 1. Apply the triangle inequality to (4.8.5) and


use Re(l1) ≥ 0 to conclude

(4.8.6) eq(a)|a|Re(l1) + 1 ≥ e(q(a)|a|+k)Re(l1) − 1.

Note that we have |a| ≥ 1, or else we would be back in case (i). Now (4.8.6) and q(a) ≥ 1imply

(4.8.7) eRe(l1) − 2e−Re(l1) − 1 ≤ 0.

The left-hand side of (4.8.7) increases in Re(l1). Substitute log 2 for Re(l1) to getthe bound Re(l1) ≤ log 2. Now assume k ≤ −1; then similar arguments as aboveinvolving the triangle inequality and this time |b| ≥ 1 also lead to (4.8.7) and thus againRe(l1) ≤ log 2. �

CHAPTER 5

More on heights and some algebraic geometry

The purpose of this chapter is to review some facts from algebraic geometry andtheir application to heights. Furthermore, we give a glimpse of some results concerningheights which will be used in the second part of the thesis. The two subsequent chapterswill make use of the concepts introduced here. Therefore we do not strive for utmostgenerality; we present the material in a form useful for our applications. Also, we giveno details of the constructions and almost no proofs.

1. Some preliminaries

By a variety we mean a Zariski open subset of a possibly reducible projective al-gebraic variety defined over a field K. For simplicity we assume throughout the wholechapter that K is algebraically closed and of characteristic zero unless stated otherwise.

Throughout this section X and Y will denote irreducible varieties defined over K.All morphisms are considered to be defined over this field. By abuse of notation we willoften identify varieties with their set of K-rational points.

We start off with the Fibre Dimension Theorem, a result we will often apply in thenext chapters.

Theorem 5.1. Let f : X → Y be a morphism of varieties. Then for p ∈ X allirreducible components of f−1(f(p)) have dimension greater or equal to dim X −dim Y .Furthermore, there exists a Zariski open, dense subset V ⊂ Y such that if q ∈ V thenall irreducible components of f−1(q) have dimension dim X − dim Y .

Proof. Follows easily from the first theorem on page 228 of [Dan94]. �

The next results is the Theorem on Semi-continuity of Fibre Dimension; it will beused in chapter 6.

Theorem 5.2. Let f : X → Y be a morphism of varieties and say k ∈ Z. The set

{p ∈ X; dimp f−1(f(p)) ≥ k}is Zariski closed in X.

Proof. This is the second theorem on page 228 of [Dan94]. �

The following result will also be useful:

Theorem 5.3. Let f : X → Y be a dominant morphism of varieties. Then f(X)contains an Zariski open, dense subset of Y .

65


Proof. Follows from the theorem on page 219 of [Dan94]. �

2. Chow forms and higher dimensional heights

We continue using the notation of the previous section but with the exception thatwe now assume X ⊂ Pn to be an irreducible closed projective variety defined over K.

The goal of this section is to define the height of a variety defined over Q. The mostnaive approach would be to define this height somehow in terms of the height of definingequations for X. A hypersurface, i.e. a subvariety of codimension 1, is the zero set ofone homogeneous polynomial unique up to multiplication by a non-zero scalar. Ournaive approach works in this situation because of the product formula. Unfortunatelyif 0 ≤ dim X ≤ n − 2 there is no unique system of polynomials which define X. Thesolution is to take a suitable height of the Chow form of X. We follow Philippon’sdefinition in [Phi95]. We begin by defining the Chow form of X.

Let dim X = r. For 0 ≤ i ≤ r, let Ui = (Ui0, . . . , Uin) where the Uij are inde-pendent variables. By Proposition 2.2 on page 99 and the discussion on pages 100-101of [GKZ94] there exist d ∈ N and FX ∈ K[Uij ; 0 ≤ i ≤ r, 0 ≤ j ≤ n] with FX

multi-homogeneous of degree d in each Ui such that

FX(u0, . . . , ur) = 0 if and only if {p ∈ X; u0(p) = · · · = ur(p) = 0} 6= ∅.

Here ui ∈ Kn+1 are identified with homogeneous polynomials of degree 1. The form FX

is called Chow form or sometimes also Cayley form or Elimination form. The integerd is just the geometric degree deg(X) which equals the cardinality of the intersectionof X with a generic linear subvariety of Pn of dimension n− r. The Chow form FX isdetermined uniquely up to multiplication by a non-zero scalar. A converse statementholds: the Chow form FX determines a set of homogeneous polynomials whose set ofcommon zeros is precisely X. Indeed, for 0 ≤ i ≤ r let s(i) = (s(i)

jk ) denote skew-

symmetric (n + 1) × (n + 1) matrices with coefficients s(i)jk (0 ≤ j, k ≤ n) which are

algebraically independent for j < k. Then

(5.2.1) FX((X0, . . . , Xn)s(0), . . . , (X0, . . . , Xn)s(r))

considered as a polynomial in the variables X0, . . . , Xn and coefficients in the fieldK(s(i)

jk ) vanishes precisely on X. Then (5.2.1) provides us with a finite number ofhomogeneous polynomials of degree d(r + 1) whose set of common zeros in Pn is X.

For the rest of the section we assume that K = Q. The coefficients of FX arecontained in a number field F . The height of X is essentially the height of the polyno-mial FX but with a different norm at the infinite places. This deviation is motivatedby Arakelov theory and the definition of the so-called Faltings height in (3.1.2.2) of[BGS94]. The norms are chosen such that the resulting height equals Philippon’sheight in [Phi95].

More precisely, let v ∈ MF . If v is a finite place, then we set ||FX ||v = |FX |v, where| · |v is defined as in section 2, chapter 1. If v is an infinite place and f is the image of

5. MORE ON HEIGHTS AND SOME ALGEBRAIC GEOMETRY 67

FX under an embedding F → C corresponding to v, we define

log ||FX ||v =∫

Sr+1

log |f(u)|σr+1 + (r + 1) deg(X)n∑

i=1

12i

,

where S ⊂ Cn+1 is the unit sphere and σ is the rotation invariant measure on S of totalmeasure 1. We define the height of X as

hV (X) =1

[F : Q]

∑v∈MF

dv log ||FX ||v,

where dv are the local degrees defined in chapter 1 section 1. This quantity does notdepend on the number field F containing the coefficients of FX .

At this point we must warn the reader about a possible ambiguity: a point p ∈Pn(Q) already has a well-defined height from chapter 1 which does not necessarilyequal the height hV ({p}) of the corresponding variety. For this reason, we use twodifferent symbols to distinguish the two heights.

There is a natural embedding ι : Gnm ↪→ Pn given by taking a point (p1, . . . , pn)

to [1 : p1 : · · · : pn]. This embedding will be used throughout the rest of the thesis.Having defined the degree and height of a subvariety of Pn it makes sense to speak of thedegree and height of a subvariety of Gn

m using this embedding. Concretely, if Z ⊂ Gnm is

Zariski dense in an irreducible subvariety defined over Q, we define hV (Z) = hV (ι(Z))and deg(Z) = deg(ι(Z)).

Say Y ⊂ Pn is an irreducible projective variety. Bezout’s Theorem provides an upperbound for the degrees of the irreducible components of the set theoretic intersectionX ∩ Y .

Theorem 5.4. Let Z1, . . . , Zg be the distinct irreducible components of X ∩Y , theng∑

i=1

deg(Zi) ≤ deg(X) deg(Y ).

Proof. See [Ful84] example 8.4.6 on page 148 or Main Theorem of [Vog84] onpage 60. �

From the point of view of Arakelov theory the height of a variety should be un-derstood as an arithmetic version of the degree. The arithmetic analog of Bezout’sTheorem is then naturally called the Arithmetic Bezout Theorem. If Y is defined overQ it bounds the height of the irreducible components in X ∩ Y from above in terms ofheights and degrees of X and Y :

Theorem 5.5. Let Z1, . . . , Zg be the distinct irreducible components of X ∩Y , theng∑

i=1

hV (Zi) ≤ deg(X)hV (Y ) + deg(Y )hV (X) + cdeg(X) deg(Y ),

where c is a positive effective constant which depends only on n.


Proof. See [Phi95] Theorem 3. �

It is clear that Theorems 5.4 and 5.5 hold if X and Y are subvarieties of Gnm. We will

apply Bezout’s Theorem in chapter 7 and the Arithmetic Bezout Theorem in chapter 6.

3. Normalized height and essential minimum of a variety

All varieties in this section are defined over Q. Let X ⊂ Gnm denote an irreducible

closed subvariety. We use h to denote the absolute logarithmic Weil height restrictedfrom Pn(Q) to Gn

m(Q).In [Phi95] Philippon described how to normalize the height of a subvariety of an

abelian variety. This construction is a generalization to higher dimension of Tate’snormalization of the height of a point on an abelian variety. In [DP99] David andPhilippon did the same for subvarieties of Gn

m. For an integer m ∈ Z, let [m] : Gnm →

Gnm denote the homomorphism which takes p to pm. In [DP99] the authors showed

that the limit

(5.3.1) h(X) = limm→∞

hV ([m]X) deg(X)m deg([m]X)

exists. Its value h(X) is called the normalized height of X. In Proposition 2.1 of [DP99]it is showed that

(5.3.2) |h(X)− hV (X)| ≤ 72(dim X + 1) deg(X) log(n + 1).

Furthermore, if p ∈ Gnm(Q), then h({p}) = h(p) where the left-hand side is the absolute

logarithmic Weil height. In view of (5.3.2) we conclude |hV ({p})− h(p)| ≤ 72 log(n + 1).

If X is the translate of an algebraic subgroup by a torsion point, then by Proposition2.1 of [DP99] we have h(X) = 0. We will see a converse statement further down.

We define the essential minimum of X as

µess(X) = supZ(X

inf{h(p); p ∈ (X\Z)(Q)},

here Z runs over all proper Zariski closed subsets of X. Equivalently one could takethe supremum over all Zariski closed subsets of X with codimension 1 in X. Anotherequivalent definition is

µess(X) = inf{δ ∈ R; {p ∈ X(Q); h(p) ≤ δ} is Zariski dense in X}.

For example if X = {p} is a point, then µess(X) = h(p). The case where X is a curveis more interesting. Then for any δ < µess(X) there are at most finitely many pointsp ∈ X(Q) with h(p) ≤ δ.

If X is the translate of an algebraic subgroup of Gnm by a torsion point then

µess(X) = 0. Hence both normalized height and essential minimum of such an Xvanish. The following theorem of Zhang explains this observation:

5. MORE ON HEIGHTS AND SOME ALGEBRAIC GEOMETRY 69

Theorem 5.6 (Zhang). We have the inequality

(5.3.3) µess(X) ≤ h(X)deg(X)

≤ (1 + dim X)µess(X).

Proof. Follows easily from Theorem 5.2 in [Zha95a] and (5.3.1). Compare alsoTheorem 1.10 of [Zha95b]. �

In fact Zhang proved a stronger lower bound: the left side of (5.3.3) can be replacedby the sum of µess(X) with the higher successive minima. The higher successive minimaare non-negative; we will not define or use them here though.

It is an important problem to determine all X with µess(X) = 0 or equivalentlyh(X) = 0. The Bogomolov Conjecture for Gn

m states that if µess(X) = 0 then X is thetranslate of an algebraic subgroup by a torsion point. Bogomolov originally conjecturedthat if C is a non-singular projective curve of genus at least 2 embedded in its Jacobian,then there exists an ε > 0 such that there are only finitely many points in C(Q) withNeron-Tate height at most ε. This conjecture was proved by Ullmo in [Ull98] and amore general version was proved by Zhang in [Zha98].

In Theorem 6.2 of [Zha95a] Zhang proved Bogomolov’s Conjecture in the algebraictorus. Zhang’s Theorem 6.2 in [Zha95a] together with Theorem 5.6 above can be seenas a higher dimension generalization of Kronecker’s Theorem. Zhang also showed thatif X∗ is X deprived of all its subvarieties that are translates of algebraic subgroups bytorsion points, then X∗ ⊂ X is Zariski open and infp∈X∗(Q) h(p) > 0. Bombieri andZannier gave an elementary and effective proof of these two statements in [BZ95]. Theyused a slightly different height in their paper which can easily be compared with ourheight. Finding lower bounds for the infimum (if X∗ is non-empty) is often called theLehmer Problem. Lower bounds involve the field of definition of X, deg(X), and n. Seefor example Corollary 1.3 in [AD01].

If we assume that X is not equal to the translate of a proper algebraic subgroup ofGn

m, then in [AD03] Amoroso and David proved a lower bound for µess(X) which onlydepends on deg(X) and n but not on the field of definition of X. The dependence inthe degree is best-possible up to a logarithmic factor. Actually, instead of the degreeAmoroso and David used the more subtle obstruction index. If V is a proper subvarietyof Pn the obstruction index is defined as

ω(V ) = infV⊂Z(Pn

{deg(Z)},

here Z runs over all Zariski closed subsets of Pn that contain V and whose irreduciblecomponents have dimension n− 1. As Amoroso and David pointed out in their article,a result of Chardin implies

(5.3.4) ω(V ) ≤ n deg(V )1/ codim V .

Using the embedding ι : Gnm ↪→ Pn we may consider the obstruction index of

subvarieties X ⊂ Gnm.


Theorem 5.7 (Amoroso, David). Assume X ( Gnm is a proper irreducible closed

subvariety of codimension k defined over Q. If X is not contained in the translate of aproper algebraic subgroup of Gn

m, then

µess(X) ≥ c

ω(X)(log(3ω(X)))−λ(k)

where c > 0 depends only on n and λ(k) = (9(3k)k+1)k.

Proof. This is Theorem 1.4 of [AD03]. �

If X is as in Theorem 5.7 then (5.3.4) implies

(5.3.5) µess(X) ≥ c′

deg(X)1/ codim X(log(3 deg(X)))−λ(k),

where c′ > 0 depends only on n.Theorem 5.7 will be applied in chapter 7 to obtain the following result: if X ⊂ Gn

m

is an irreducible curve not contained in the translate of an algebraic subgroup then thereare at most finitely many points in X(Q) that lie in an algebraic subgroup of dimensionn− 2 “up to an ε”. In the proof of this result it will be essential that the lower boundin (5.3.5) is best possible up to a power of a logarithm.

CHAPTER 6

Intersecting varieties with small subgroups

Let X ⊂ Gnm be an irreducible closed subvariety. In this chapter we will bound

the height of algebraic points on X that are in a certain sense close to the union of allalgebraic subgroup of Gn

m of dimension m < n/dim X. These results give a qualitativegeneralization of results from previous chapters. The bounds we obtain are effective andwill be expressed as functions of the height of X, the degree of X, and n. We will usethese height bounds to derive finiteness results if the points in question actually lie onthe union of all algebraic subgroup of dimension m. In chapter 7 we will prove finitenessresults for points close to such a subgroup.

1. Introduction

Unless stated otherwise, all varieties in this chapter are assume to be defined over Q.Fix an irreducible closed subvariety X ⊂ Gn

m. In this chapter and the next a coset is thetranslate of an algebraic subgroup of Gn

m. We do not require a coset to be irreducible.Let H be a subset of Gn

m(Q) and let ε ≥ 0, we define the “cone” around H:

C(H, ε) = {ab; a ∈ H, b ∈ Gnm(Q), h(b) ≤ ε(1 + h(a))},

here h(·) is the absolute logarithmic Weil height on Gnm(Q) defined in chapter 1.

Let Xo be the complement in X of the union of all positive dimensional cosetscontained in X. The work of Bombieri and Zannier (Theorem 1, [BZ95]) shows thatXo is Zariski open in X. Let Γ be a finitely generated subgroup of Gn

m(Q) and let Γ bethe group of p ∈ Gn

m(Q) such that pk ∈ Γ for a positive integer k. Evertse showed inTheorem 1.7 of [Eve02] that Xo ∩ C(Γ, ε) is finite for some positive ε. The definitionof Xo makes sense if Gn

m is replaced by any semi-abelian variety A. One can define areasonable height on A(Q) if A is defined over Q. In this context Poonen ([Poo99])proved that if A is isogenous to the product of an abelian variety with Gn

m (n = 0 isallowed) then there exists a positive ε with the following property: there are at mostfinitely many points p = ab ∈ Xo where a ∈ Γ and where b has height at most ε. In factPoonen’s Corollary 9 is slightly stronger. In [Rem03] Theorem 1.2 Remond proves thefiniteness of Xo ∩ C(Γ, ε) for any semi-abelian variety A defined over Q and for someε > 0.

For ε = 0 we describe a result by Bombieri, Masser, and Zannier in [BMZ99]which concerns the intersection of X with the union of all algebraic subgroups of fixeddimension. When 0 ≤ m ≤ n is an integer, we define Hn

m to be the union of all algebraicsubgroups of Gn

m with dimension less or equal to m. If not stated otherwise, we usually

71


identify Hnm with the set of its algebraic points. To avoid trivialities we set Hn

m = ∅for negative m. In [BMZ99] the authors showed that if X ⊂ Gn

m is an irreduciblealgebraic curve not contained in a proper coset, then X ∩Hn

m is a set of bounded heightfor m = n− 1 and finite for m = n− 2.

Motivated by the results described in the previous two paragraphs we pose theproblem of finding boundedness of height results for the intersection Xoa∩C(Hn

m, ε) fora small ε > 0 and appropriate m. Here X is not necessarily a curve and Xoa ⊂ X willbe defined further down.

In some instances height bounds for Xoa ∩C(Hnm, ε) imply the finiteness of this set

as we will see later on. We will discuss finiteness results with ε = 0 here and with ε > 0in chapter 7. By Kronecker’s Theorem the set C(Hn

m, 0) equals Hnm. Hence Bombieri,

Masser, and Zannier’s finiteness result in [BMZ99] covers the case ε = 0 for curves notcontained in a proper coset.

We start off with a short review of what else is known about intersections X ∩C(Hn

m, ε) from the point of view of boundedness of height and finiteness.If we have m = 0 thenHn

0 is the set of torsion points of Gnm by Kronecker’s Theorem.

Certainly the height is bounded on X ∩ Hn0 , but finiteness questions are already inter-

esting: one is lead to the study of torsion points contained in X. For example, a specialcase of a result of Laurent in [Lau84] shows that the intersection X ∩Hn

0 is containedin a finite union of translates of algebraic subgroups by torsion points. Furthermore,these translates are contained in X.

Next, if we allow ε to be positive but still with m = 0, then C(Hn0 , ε) is just the

set of algebraic points Gnm with height at most ε. So again bounding the height on

X ∩C(Hn0 , ε) is trivial. Finiteness (and non-density) questions of this set are related to

the Bogomolov Conjecture already discussed to some extent in section 3 of chapter 5.Up to now most work concerning the intersection of varieties with C(Hn

m, ε) wherepositive dimensional algebraic subgroups are involved assumed ε = 0. Hence we continueour review considering only the case ε = 0.

For X of any dimension, Bombieri and Zannier showed in Theorem 1 ([Zan00] page524) that Xo∩Hn

1 is a set of bounded height. The subgroup dimension in this theorem isnot believed to be best-possible for dim X ≤ n−2. This belief is supported by [BMZ99]where the subgroups have dimension n−1, i.e. equal to the codimension of the variety inquestion. For 2 ≤ dim X ≤ n− 2 only one boundedness of height result is known wherem > 1: if X is a plane, then in the preprint [BMZ04] Bombieri, Masser, and Zannierproved that Xoa ∩ Hn

n−2 has bounded height and Xoa ∩ Hnn−3 is finite. Furthermore,

they proved that for planes, the still undefined set Xoa ⊂ X is Zariski open.In [BMZ06b] Bombieri, Masser, and Zannier proved that Xta ∩ Hn

1 is finite ifdim X = n− 2 and in this case Xta is Zariski open in X. The definition of Xta will alsobe given below.

As already mentioned, if X is a curve which is not contained in a proper coset, thenX ∩ Hn

n−2 is finite. Say now that X is contained in a proper coset, then Bombieri,Masser, and Zannier showed that the height on X ∩ Hn

n−1 is necessarily unbounded.Nevertheless it is conjectured in [BMZ99] that X ∩Hn

n−2 is finite if X is allowed to be

6. INTERSECTING VARIETIES WITH SMALL SUBGROUPS 73

contained in a proper coset but not in a proper algebraic subgroup. Recently, Maurinhas announced a proof of this conjecture in [Mau06].

We will prove the following simple proposition:

Proposition 6.1. Let X ⊂ Gnm be an irreducible closed subvariety defined over Q

of dimension r with 1 ≤ r ≤ n and let U ⊂ X be Zariski open and dense. Then the setU ∩Hn

n−r+1 has unbounded height and U ∩Hnn−r is infinite.

Therefore to prove a reasonable boundedness of height result on intersections ofX with Hn

m one needs to assume m ≤ n − dim X. And to prove finiteness resultsone needs m ≤ n − dim X − 1. There is a series of conjectures stated by Bombieri,Masser, and Zannier which state that one can take m to be equal to these upper boundsand expect boundedness of height or finiteness respectively, at least when restricting togeometrically motivated subsets Xoa of X which we now finally define.

An irreducible closed subvariety Y ⊂ X is called X-anomalous if there exists a cosetH ⊂ Gn

m such that Y ⊂ H and

(6.1.1) dim Y ≥ max{1,dim X + dim H − n + 1}.

If Y ⊂ X is X-anomalous and there can be no confusion regarding X we will simply callY anomalous. We define Xoa to be X deprived of all its anomalous subvarieties. If H isan algebraic subgroup we call Y X-torsion anomalous, or just torsion anomalous. Wedefine Xta to be X deprived of all its torsion anomalous subvarieties. These definitionscontinue to make sense when all varieties in question are defined over C.

Informally speaking, a positive dimensional Y ⊂ X is anomalous if it is containedin a component of the intersection of X with a coset whose dimension is smaller thanexpected.

We present some special cases where Xoa and Xta can easily be determined: if X isa curve, then Xoa = X if X is not contained in a proper coset, and Xoa = ∅ otherwise.Similarly, Xta = X if X is not contained in a proper algebraic subgroup, and Xta = ∅otherwise. If X is a hypersurface, i.e. if dim X = n − 1, then it is easy to see thatXoa = Xo.

Conjecture 6.1. Let X ⊂ Gnm be an irreducible closed subvariety defined over C.

Then Xoa ⊂ X is Zariski open.

In the work [BMZ06b], Bombieri, Masser, and Zannier proved Conjecture 6.1 andeven a “Structure Theorem” for anomalous subvarieties.

The conjectures of Bombieri, Masser, and Zannier are:

Conjecture 6.2. (Bombieri, Masser, Zannier) Let X ⊂ Gnm be an irreducible

closed subvariety defined over Q. Then Xoa ∩Hnn−dim X is a set of bounded height.

Conjecture 6.3. (Bombieri, Masser, Zannier) Let X ⊂ Gnm be an irreducible

closed subvariety defined over C. Then Xta ⊂ X is Zariski open and furthermoreXta ∩Hn

n−dim X−1 is finite.


In Conjecture 6.3 one identifies Hnm with the set of its complex points.

By the survey of known results above, Conjecture 6.2 has been proved for curves,hypersurfaces, and planes.

Pink and Zilber independently stated conjectures concerning general semi-abelianvarieties which are related to the conjectures above. In [Pin05b], Pink conjectured:

Conjecture 6.4 (Pink, Zilber). Let A be a semi-abelian variety defined over C andlet X ⊂ A be an irreducible closed subvariety also defined over C which is not containedin a proper algebraic subgroup of A. Then the set of points in X(C) that are containedin an algebraic subgroup of A of codimension greater or equal to dim X +1 is not Zariskidense in X.

In [Zil02] Zilber stated a conjecture which implies Conjecture 6.4.Theorem 6.1, the main result of this chapter, goes in the direction of Conjecture

6.2 but with positive ε. The drawback is that the subgroups involved have dimensionstrictly less than n/ dim X. This strong restriction on the subgroup dimension allowsus to prove Conjecture 6.2 only in some special cases.

To formulate Theorem 6.1 we need a refined version of Xoa: let t be an integer with0 ≤ t ≤ n, we define Xoa,t to be X deprived of all its closed irreducible subvarieties Ythat are contained in a coset H satisfying (6.1.1) and dim H ≤ t. Clearly we have

(6.1.2) X = Xoa,0 ⊃ Xoa,1 ⊃ · · · ⊃ Xoa,n = Xoa.

And even

(6.1.3) Xoa,n−dimX = Xoa.

Indeed by (6.1.2) it suffices to prove that if p ∈ Y ⊂ pH where Y ⊂ X and H is analgebraic subgroup satisfying (6.1.1), then p /∈ Xoa,n−dimX . If dim H ≤ n − dim X,then we are done; hence say dim H > n − dim X. We will see in Proposition 6.2 thatthere exist linearly independent u1, . . . , un−dim H ∈ Zn such that H = {x ∈ Gn

m, xu1 =· · · = xun−dim H = 1}. Here notation introduced in section 2 is used. We may choosev1, . . . , vdim H+dim X−n ∈ Zn such that the ui, vi are linearly independent. By Proposition6.2 the vi define an algebraic subgroup H ′ ⊂ Gn

m with dim H ′ = 2n − dim H − dim Xand dim H∩H ′ = n−dim X. Let Y ′ be an irreducible component of Y ∩pH ′ containingp. By [Ful84] (§8.2, page 137) we have dim Y ′ ≥ dim Y +dim H ′−n and we use (6.1.1)to conclude dim Y ′ ≥ 1. Furthermore, Y ′ ⊂ p(H ∩H ′) where the right side is a coset ofdimension n− dim X. We conclude p /∈ Xoa,n−dimX . Therefore (6.1.3) is established.

We are now ready to state Theorem 6.1.

Theorem 6.1. Let X ⊂ Gnm be an irreducible closed subvariety defined over Q. Let

s be an integer with dim X ≤ s ≤ n and let m be an integer with m · s < n. Then thereexists ε > 0 which depends only on deg(X) and n such that if p ∈ Xoa,n−s ∩ C(Hn

m, ε)then

h(p) ≤ c(n) · deg(X)ms

n−ms (deg(X) + hV (X)).

The constant c(n) is effective and depends only on n.


We recall that degrees and heights of subvarieties of Gnm are defined in chapter 5.

It is tempting to take s = dim X in Theorem 6.1, since then one can take m as largeas possible and also Xoa,n−dimX = Xoa. If X is a curve or a hypersurface, then wemay choose m = n − 1 or m = 1 respectively. In these two cases the theorem impliesConjecture 6.2 “with an ε” and with explicit height bounds. Although Conjecture 6.2 hasalready been proved for such X, no explicit height bounds have appeared in literature.

The proof of Theorem 6.1 does not use the now proven fact (cf. [BMZ06b]) thatXoa ⊂ X is Zariski open. In fact if s ≥ dim X the proof shows that the height isbounded above on U ∩ C(Hn

m, ε) for some non-empty Zariski open set U ⊂ X withXoa,n−s ⊂ U as soon as the following hypothesis on X is satisfied.

(6.1.4)For any surjective homomorphism of algebraic groupsϕ : Gn

m → Gsm one has dim ϕ(X) = dim X.

In Lemma 6.4 we will show that (6.1.4) follows from Xoa,n−s 6= ∅.If X 6= Gn

m then taking s = n − 1 also has interesting consequences. Indeed it isnot difficult to see that Xoa,1 = Xo. Now Theorem 6.1 with m = 1 gives an explicitheight bound for the set Xo ∩C(Hn

1 , ε). We have therefore recovered an explicit versionof Bombieri and Zannier’s Theorem ([Zan00] Theorem 1, page 523) mentioned above.Their theorem also holds for X = Gn

m, but then of course Xo = ∅.In [BMZ04] Bombieri, Masser, and Zannier show the following theorem:

Theorem 6.2. (Bombieri, Masser, Zannier) Let X ⊂ Gnm be an irreducible closed

subvariety defined over Q of dimension r. If B ∈ R, then

{p ∈ Xta ∩Hnn−r−1; h(p) ≤ B}

is finite.

If we combine Bombieri, Masser, and Zannier’s result from [BMZ04] with Theorem6.1 we get a finiteness result with ε = 0.

Corollary 6.1. Let X ⊂ Gnm be an irreducible closed subvariety defined over Q of

dimension r ≥ 1 and let m be an integer with m < min{n/r, n− r}. Then Xoa ∩Hnm is

finite.

The proof is an immediate consequence of Theorems 6.1 (with s = dim X) and 6.2,and the fact that Xoa ⊂ Xta.

For example, the previous corollary implies the finiteness statement of Conjecture6.3 in the cases of curves (r = 1), hypersurfaces (r = n − 1), and also r = n − 2 butalways with Xta replaced by Xoa and if everything is defined over Q.

There is one further situation where Corollary 6.1 implies finiteness with the correctsubgroup dimension: namely surfaces in G5

m.

Corollary 6.2. Let X ⊂ G5m be an irreducible closed algebraic surface defined over

Q. Then Xoa ∩H52 is finite.

Of course this corollary follows from Corollary 6.1 by taking n = 5, r = 2, andm = 2.


We remark that we are showing the finiteness of Xoa∩H52 and not of the potentially

larger set Xta ∩H52 which appears in Conjecture 6.3.

As we will see, proving a version of Theorem 6.1 with ε = 0 is not essentially simplerthan the proof of Theorem 6.1 itself. Hence in the problem of bounding the height weget the ε for free. Unfortunately the same cannot be said about the problem of provingfiniteness in the style of Theorem 6.2. In fact we reserve the whole next chapter to provea version of Theorem 6.2 “with an ε”. Although the subgroup dimension m will oftenbe less than n− r − 1.

2. Algebraic subgroups of Gnm

We let c1, . . . , c6 denote positive constants which depend only on n.As all our work is done in the torus Gn

m we introduce some notation which easeswork in this and the next chapter. Let K be a field. Let p = (p1, . . . , pn) ∈ Gn

m(K)and u = (α1, . . . , αn) ∈ Zn, we set pu = pα1

1 · · · pαnn . Say u1, . . . , um ∈ Zn. If U is an

n×m matrix with columns u1, . . . , um ∈ Zn then we set pU = (pu1 , . . . , pum) ∈ Gmm(K).

For q ∈ Gnm(K) we have (pq)U = pUqU . If V is a matrix with m rows and integer

coefficients, then (pU )V = pUV . If K = R and all pi are positive we will also allowexponent vectors or matrices with rational numbers as entries.

Let p = (p1, . . . , pn) ∈ Gnm(Q) and say U is as above. By elementary height inequal-

ities we have

(6.2.1) h(pU ) ≤ m

n∑j=1

xjh(pj),

where xj is the maximum of the absolute values of the elements of the jth row of U .Finally we define the morphism of algebraic groups ϕ(u1,...,um) : Gn

m → Gmm by

sending p to (pu1 , . . . , pum).For u ∈ Rn |u| will always denote the euclidean norm of u. Furthermore, if L ∈

R[X1, . . . , Xn] is a linear form, then |L| will denote the euclidean norm of the coefficientvector of L.

Let Λ ⊂ Zn be a subgroup. Then we define

H(Λ) = {p ∈ Gnm; pu = 1 for all u ∈ Λ}.

It is clear that H(Λ) is an algebraic subgroup of Gnm. We define detΛ to be the deter-

minant of the subgroup Λ considered as a lattice in Rn.Algebraic subgroups of Gn

m and subgroups of Zn are closely related:

Proposition 6.2. Let Λ ⊂ Zn be a subgroup of rank r, then dimH(Λ) = n− r and

(6.2.2) deg(H(Λ)) ≤ c1 det Λ.

Furthermore, for any algebraic subgroup H ⊂ Gnm there exists a subgroup Λ ⊂ Zn with

H = H(Λ).

Proof. The equality dimH(Λ) = n− r follows from Proposition 3.2.7 in [BG06].The last statement of the assertion is Corollary 3.2.15 of [BG06].


Inequality (6.2.2) can be proved as follows: by Minkowski’s Second Theorem thereexist linearly independent u1, . . . , ur ∈ Λ with |u1| · · · |ur| ≤ cdet Λ, here c dependsonly on n. Say the ui generate the subgroup Λ′ ⊂ Zn. Then by the first part of theproposition H(Λ) and H(Λ′) have equal dimension. Since H(Λ) ⊂ H(Λ′) the irreduciblecomponents of H(Λ) are irreducible components of H(Λ′). Now xu1 = · · · = xur = 1are defining equations for H(Λ′). By multiplying these with suitable monomials we getpolynomial equations. Inequality (6.2.2) then follows from Bezout’s Theorem (Theorem5.4). �

3. Some geometry of numbers

We start off with a result very much in the spirit of an argument given in thealternative proof of Theorem 1 in [BMZ99].

Lemma 6.1. Let 1 ≤ m ≤ n and let a ∈ Hnm. Then there exist linear forms

L1, . . . , Lm ∈ R[X1, . . . , Xn] such that |Lj | ≤ 1 and

h(au) ≤ c2 max1≤j≤m

{|Lj(u)|}h(a)

for all u ∈ Zn.

Proof. Let a = (a1, . . . , an) ∈ Hnm; then by Proposition 6.2 we may assume that

the ai lie in a finitely generated subgroup of Q∗ of rank m. By a result of Schlickewei(Theorem 1.1 of [Sch97]) there exist multiplicatively independent elements g1, . . . , gm ∈Q∗ and roots of unity ζ1, . . . , ζn such that

(6.3.1) ai = ζigvi11 · · · gvim

m for some vij ∈ Z

and

(6.3.2) h(gb11 · · · gbm

m ) ≥ c(|b1|h(g1) + · · ·+ |bm|h(gm))

for all (b1, . . . , bm) ∈ Zm; here c > 0 depends only on n.We define A = maxi,j{|vij |h(gj)}. The assertion of the lemma obviously holds if all

ai are roots of unity. So let us assume that at least one vij is non-zero; then A > 0. For1 ≤ j ≤ m we define the linear forms

(6.3.3) Lj = v1jX1 + · · ·+ vnjXn and Lj =h(gj)√

nALj .

Clearly we have |Lj | ≤ 1.Let u ∈ Zn, then (6.3.1) and (6.3.3) imply

au = ξgL1(u)1 · · · gLm(u)

m


for some root of unity ξ. We apply standard height properties and (6.3.3) to conclude

h(au) = h(gL1(u)1 · · · gLm(u)

m )

≤ |L1(u)|h(g1) + · · ·+ |Lm(u)|h(gm)

=√

nA(|L1(u)|+ · · ·+ |Lm(u)|)≤ m

√n max

1≤j≤m{|Lj(u)|}A.(6.3.4)

We choose i0 and j0 such that A = |vi0j0 |h(gj0). Then (6.3.1) and (6.3.2) imply

h(a) ≥ h(ai0) ≥ c|vi0j0 |h(gj0) = cA.

We insert this inequality into (6.3.4) to complete the proof. �

Now we approximate zeros of linear forms with real coefficients by integral vectorswith controlled coefficients. The next lemma will be used in this chapter and also inchapter 7. The proof uses Minkowski’s Second Theorem.

Lemma 6.2. Let 1 ≤ m ≤ n and let L1, . . . , Lm ∈ R[X1, . . . , Xn] be linear formswith |Lj | ≤ 1. If ρ ≥ 1, there exist λ1, . . . , λn with 0 < λ1 ≤ · · · ≤ λn and linearlyindependent u1, . . . , un ∈ Zn such that

(6.3.5) |uk| ≤ λk, |Lj(uk)| ≤ ρ−1λk, and λ1 · · ·λn ≤ c3ρm.

Proof. We set Λ ⊂ Rm+n to be the rank n lattice generated by the columns of the(m + n)× n-matrix

A =

ρL1

...ρLm

E

where E is the n × n unit matrix and the Li are identified with the coefficient vectorsin Rn. By the Cauchy-Binet formula we have det Λ = (detAtA)1/2 = (

∑A′(detA′)2)1/2

where the sum ranges over all n × n minors A′ of A. By Hadamard’s inequality wededuce |det A′| ≤ ρm and so

det Λ ≤(

n + m

n

)1/2

ρm.

Let Q = {x ∈ Rm+n; |x| ≤ 1} be the unit ball and V = Λ ⊗R ⊂ Rm+n. Then V

is an n-dimensional vector space and vol(Q ∩ V ) = πn/2/Γ(1 + n/2). Here vol is then-dimensional Lebesgue volume on V and Γ is the usual Gamma function. In particularvol(Q ∩ V ) depends only on n and not on V .

Let λ1, . . . , λn be the successive minima of Λ with respect to the convex, symmetric,and compact set Q ∩ V . Minkowski’s Second Theorem implies

(6.3.6) λ1 · · ·λn ≤ 2n det Λvol(Q ∩ V )

≤ c3ρm


with c3 = 2n(

n+mn

)1/2 Γ(1 + n/2)/πn/2. By definition there exist

(6.3.7) vk ∈ (λkQ) ∩ Λ with v1, . . . , vn linearly independent.

For 1 ≤ k ≤ n there are uk ∈ Zn with

(6.3.8) vk = (ρL1(uk), . . . , ρLm(uk), uk).

Clearly the u1, . . . , un are also linearly independent. The first two inequalities in (6.3.5)follow from (6.3.7) and (6.3.8). The last one is just (6.3.6). �

Lemma 6.3. Let 1 ≤ m ≤ n and let L1, . . . , Lm ∈ R[X1, . . . , Xn] be linear formswith |Lj | ≤ 1. If T ≥ 1, then for any integer s with 1 ≤ s ≤ n there exist u1, . . . , us ∈ Zn

linearly independent such that |u1| · · · |us| ≤ T and

|u1| · · · |us||Lj(uk)||uk|

≤ c4T1− n

ms for 1 ≤ j ≤ m and 1 ≤ k ≤ s.

Proof. Say c3 is the constant from Lemma 6.2. If T < cs/n3 , then c3 ≥ 1 and

T 1− nsm ≥ T 1−n

s ≥ T−ns > c−1

3 .

In this case it suffices to take for u1, . . . , us any distinct standard basis elements of Rn

and c4 = c3.So let us assume that T ≥ c

s/n3 . We set ρ = c

−1/m3 Tn/(ms) ≥ 1. Applying Lemma

6.2, we get λk and uk as in (6.3.5). Then

|u1| · · · |us| ≤ λ1 · · ·λs ≤ (λ1 · · ·λn)s/n ≤ cs/n3 ρms/n = T.

Furthermore, by (6.3.5) and the inequality above we may estimate

|u1| · · · |us||Lj(uk)||uk|

≤ ρ−1λ1 · · ·λs ≤ ρ−1T = c1/m3 T 1− n

ms

for 1 ≤ j ≤ m and 1 ≤ k ≤ s as desired. �


We start this section by pointing out a simple consequence of the assumption Xoa 6=∅. This result will not be used in this chapter.

Lemma 6.4. Let X ⊂ Gnm be an irreducible closed subvariety defined over C with

1 ≤ dim X ≤ s. Let us assume Xoa,n−s 6= ∅. Then for any surjective homomorphism ofalgebraic groups ϕ : Gn

m → Gsm one has dim ϕ(X) = dim X.

Proof. Say dim ϕ(X) < dim X, and let p ∈ X(C). The Fibre Dimension Theoremimplies that each irreducible component of ϕ|−1

X (ϕ|X(p)) has positive dimension. Eachsuch component is anomalous since it is contained in a coset of dimension n − s ≤n−dim X. One of these components contains p, therefore p /∈ Xoa,n−s . Hence Xoa,n−s =∅. �


We will now show Proposition 6.1.Proof of Proposition 6.1: Let x1, . . . , xn denote the coordinate functions on X. SinceX has dimension r we may assume without loss of generality that x1, . . . , xr are alge-braically independent over Q. Let π : Gn

m → Grm denote the projection onto the first r

coordinates. Then π(X) is Zariski dense in Grm, and so is π(U). By Theorem 5.3 π(U)

contains a Zariski open dense subset V ⊂ Grm. The torsion points of Gr

m lie Zariskidense and hence have infinite intersection with V . Now any point p ∈ U(Q) such thatπ(p) is a torsion point already lies in Hn

n−r. Therefore U ∩Hnn−r is infinite.

Let π′ : Gnm → Gr−1

m denote the projection onto the first r − 1 coordinates. Againπ′(U) lies Zariski dense and thus contains V ′ 6= ∅ which is Zariski open in Gr−1

m . NowV ′ must contain a torsion point of Gr−1

m . By the Fibre Dimension Theorem, π′|U haspositive dimensional fibres. By taking the fibre above a torsion point in V ′ we see thatU ∩Hn

n−r+1 has unbounded height. �

Before giving the main argument for the proof of Theorem 6.1 we make some pre-liminary observations.

Assume for the moment that m is an integer with 0 ≤ m ≤ n. Say p ∈ C(Hnm, ε) and

ε ≤ (2n)−1. Then p = ab with a ∈ Hnm and h(b) ≤ ε(1 + h(a)). By elementary height

properties described in chapter 1 we have h(a) = h(pb−1) ≤ h(p)+h(b−1) ≤ h(p)+nh(b).So h(a) ≤ h(p) + nε(1 + h(a)) ≤ h(p) + 1

2(1 + h(a)). We conclude that

(6.4.1) h(a) ≤ 1 + 2h(p) and h(b) ≤ 2ε(1 + h(p)).

For T ∈ R and an integer s with 1 ≤ s ≤ n we define

Φs(T ) = {ϕ(u1,...,us) : Gnm → Gs

m;

u1, . . . , us ∈ Zn linearly independent and |u1| · · · |us| ≤ T}.

Clearly Φs(T ) is a finite set.

Lemma 6.5. Let 1 ≤ m ≤ n be an integer, let p ∈ C(Hnm, ε) with ε ≤ (2n)−1, and let

T ≥ 1. Say s is an integer with 1 ≤ s ≤ n− 1; then there exists a map ϕ ∈ Φs(T ) suchthat if H is the irreducible component of ker ϕ containing 1, then the normalized heightsatisfies

(6.4.2) h(pH) ≤ c5(T 1− nms + Tε)(h(p) + 1) and deg(pH) ≤ c6T.

Proof. In this proof c′1, . . . , c′10 will denote constants which depend only on n.

We have p = ab with a ∈ Hnm and h(b) ≤ ε(1 + h(a)). By Lemmas 6.1 and 6.3 there

exist linearly independent u1, . . . , us ∈ Zn such that |u1| · · · |us| ≤ T and

(6.4.3)|u1| · · · |us|

|uk|h(auk) ≤ c′1T

1− nms h(a) ≤ 2c′1T

1− nms (1 + h(p)) for 1 ≤ k ≤ s.

The second inequality used the first part of (6.4.1). We define ϕ = ϕ(u1,...,us) ∈ Φs(T ).By elementary height inequalities (cf. chapter 1) we have h(buk) ≤ c′2|uk|h(b), hence


using the second part of (6.4.1) we get

(6.4.4)|u1| · · · |us|

|uk|h(buk) ≤ c′2|u1| · · · |us|h(b) ≤ c′2Th(b) ≤ 2c′2εT (1 + h(p))

for 1 ≤ k ≤ s. By (6.4.3) and (6.4.4), and since h(puk) ≤ h(auk) + h(buk) we conclude

(6.4.5)|u1| · · · |us|

|uk|h(puk) ≤ c′3(T

1− nms + εT )(1 + h(p)) for 1 ≤ k ≤ s.

Let U be the n× s matrix with columns u1, . . . , us. If we write uk = (u1k, . . . , unk),we may assume that

U ′ =

u11 · · · u1s...

...us1 · · · uss

is an s× s minor of U with maximal absolute determinant. We set ∆ = detU ′ and letΛ ⊂ Zn be the rank s subgroup generated by u1, . . . , us. The Cauchy-Binet formula saysthat (detΛ)2 is equal to the sum over the squares of the determinants of all s×s-minorsof U . Hence we may bound

(6.4.6) |∆| ≥ c′4 det Λ

for some positive c′4. The lattice Λ defines an algebraic subgroup kerϕ = H(Λ) ⊂ Gnm

of dimension n− s by Proposition 6.2. We let H ⊂ H(Λ) be the irreducible componentof H(Λ) that contains the unit element. If π : Gn

m → Gn−sm is the projection onto the

last n− s coordinates, then π(pH) = Gn−sm because ∆ 6= 0. Since the torsion points of

Gn−sm lie Zariski dense we conclude that the set V = π|−1

pH(torsion points of Gn−sm ) lies

Zariski dense in pH.Let w ∈ V ⊂ pH, then w = ph for some h ∈ H(Q). We define the (n−s)×s-matrix

U ′′ by

U =[

U ′

U ′′

].

Furthermore, let us set w = (w′, w′′) with w′ ∈ Gsm(Q) and w′′ ∈ Gn−s

m (Q). Thenw′U ′

w′′U ′′= wU = pUhU = pU . So w′∆ = w′U ′(#U ′) = pU(#U ′)w′′−U ′′(#U ′) where #U ′ is

the adjoint matrix of U ′. Note that the coordinates of w′′ are roots of unity. Elementaryheight properties and (6.2.1) provide

|∆|h(w) = |∆|h(w′) ≤ sh(w′∆) = sh(pU(#U ′)) ≤ c′5

s∑k=1

xkh(puk).

Here xk denotes the maximum of the absolute values of the elements of the kth row of#U ′. Linear algebra implies xk ≤ c′6

|u1|···|us||uk| . Hence with (6.4.5) we get

(6.4.7) h(w) ≤ c′7|∆|

(T 1− nms + εT )(1 + h(p)),

and this inequality holds for all w ∈ V .


Since V ⊂ pH lies Zariski dense, the right-hand side of (6.4.7) is an upper boundfor µess(pH). Zhang’s Theorem (Theorem 5.6) implies

(6.4.8) h(pH) ≤ deg(pH)(1 + dim pH)µess(pH) ≤ c′8deg(pH)|∆|

(T 1− nms + εT )(1 + h(p)).

We have

(6.4.9) deg(pH) = deg(H) ≤ deg(H(Λ)) ≤ c′9 det Λ,

where the last inequality follows from the bound in (6.2.2). We recall (6.4.6) to deduce|∆|−1 deg(pH) ≤ c′10 and so (6.4.8) implies the height bound in (6.4.2). The degreebound in (6.4.2) follows from (6.4.9) and det Λ ≤ |u1| · · · |us| ≤ T . �

We can now prove Theorem 6.1.Clearly we may assume Xoa,n−s 6= ∅. To avoid trivialities say dim X ≥ 1 and m ≥ 1.We will assume that T ≥ 1 and ε > 0 are fixed and depend only on deg(X) and

n. At the end of the proof we will see how to choose them appropriately. In this proofc′1, . . . , c

′5 will denote constants which depend only on n.

Say ϕ : Gnm → Gs

m is a surjective homomorphism of algebraic groups. We set

Zϕ = {p ∈ X; p is not an isolated point of ϕ|−1X (ϕ|X(p))}

Theorem 5.2 with k = 1 implies that Zϕ ⊂ X is Zariski closed. We claim Zϕ ⊂X\Xoa,n−s . Indeed say p ∈ Zϕ, then p is contained in an irreducible Y ⊂ X of positivedimension with ϕ(Y ) = ϕ(p). So Y is contained in the coset p ker ϕ of dimension n− s.Since dim X ≤ s we conclude p /∈ Xoa,n−s . Thus indeed Zϕ ⊂ X\Xoa,n−s . In particularZϕ 6= X by our assumption.

We define Z as the unionZ =

⋃ϕ∈Φs(T )

Zϕ.

Clearly Z ( X is Zariski closed because the union is taken over a finite set. Furthermore,Xoa,n−s ⊂ X\Z.

We proceed to show that we have bounded height on (X\Z) ∩C(Hnm, ε). Hence let

us assume p ∈ (X\Z) ∩ C(Hnm, ε) with m · s < n. We may assume ε ≤ (2n)−1. By

Lemma 6.5 there exists ϕ ∈ Φs(T ) such that if H is the irreducible component of kerϕcontaining 1, then (6.4.2) holds. Since p /∈ Zϕ we conclude that {p} is an irreduciblecomponent of the intersection X ∩pH. The Arithmetic Bezout Theorem (Theorem 5.5)implies

hV ({p}) ≤ deg(X)hV (pH) + deg(pH)hV (X) + c′1 deg(X) deg(pH).(6.4.10)

We have the bound deg(pH) ≤ c6T . To bound hV (pH) we note that by (5.3.2) inchapter 5 the inequality |h(pH) − hV (pH)| ≤ c′2 deg(pH) ≤ c′3T holds. By (6.4.2) weconclude

hV (pH) ≤ c5(T 1− nms + εT )(h(p) + 1) + c′3T.


We insert this inequality and the bound for deg(pH) into (6.4.10) to get

hV ({p}) ≤ c5 deg(X)(T 1− nms + εT )(h(p) + 1) + c6ThV (X) + c′4T deg(X).(6.4.11)

The estimate (5.3.2) lets us replace hV ({p}) by h(p) at the cost of replacing c′4 by c′5 in(6.4.11).

Without loss of generality c5 ≥ max{1, 2n/3}. As nms > 1 we choose T ≥ 1 such

thatc5T

1− nms deg(X) =

13

and then ε > 0 such thatc5Tε deg(X) =

13.

This choice implies ε ≤ (2n)−1. Inequality (6.4.11) concludes the proof after a shortcalculation. �

CHAPTER 7

A Bogomolov property modulo algebraic subgroups

Let X ⊂ Gnm be an irreducible closed subvariety defined over Q which is not con-

tained in the translate of a proper algebraic subgroup of Gnm. Given a real B we show

that for an appropriate integer m = m(dim X, n) there exists a positive ε with the fol-lowing property: the set of p ∈ X ∩ C(Hn

m, ε) with h(p) ≤ B is not Zariski dense in X.A related statement was proved with ε = 0 and optimal m = n−dim X−1 by Bombieri,Masser, and Zannier in [BMZ04] (cf. Theorem 6.2 in the previous chapter). If X is acurve, we will show that m = n− 2 suffices. This particular value of m is best possiblefor curves. We then continue by applying results from chapter 6 to deduce finitenessresults independent of B.

1. Introduction

In this chapter all varieties are considered to be defined over Q. We will work withmuch the same notation as in chapter 6, i.e. the same notion of height and the samedefinition for the set Hn

m. We also refer to chapter 6 for the motivation of the problemconsidered here.

Let 1 ≤ r ≤ n be real numbers, we define

(7.1.1) m(r, n) = n− 2r + 2−d(r(d + 2)− n) with d =[n− 1

r

],

here [x] denotes the greatest integer less or equal to x.We recall that the degree of a subvariety of Gn

m is defined using the embeddingι : Gn

m ↪→ Pn introduced in chapter 5. Recall also that a coset is the translate of analgebraic subgroup of Gn

m. The precise definition of the deprived set Xoa can be foundin chapter 6.

The main result of this chapter is:

Theorem 7.1. Let X ⊂ Gnm be an irreducible closed subvariety of dimension r ≥ 1

defined over Q. Let B ∈ R and let m be an integer with m < m(r, n).

(i) If X is not contained in a proper coset, then there exists an ε > 0 dependingonly on B, deg(X), and n such that

{p ∈ X ∩ C(Hnm, ε); h(p) ≤ B}

is not Zariski dense in X.

85


(ii) For unrestricted X there exists an ε > 0 such that

{p ∈ Xoa ∩ C(Hnm, ε); h(p) ≤ B}

is finite.

In Proposition 7.4 we provide a more precise version of part (i) of this theorem.For integers 1 ≤ r ≤ n we always have

m(r, n) > n− 2r,

hence the choice m = n−2r is always possible. In particular if r = 1, then we may takem = n− 2 in Theorem 7.1. This value of m is best possible for curves. Unfortunately ifr > n/2, then n− 2r is negative, thus uninteresting as a choice for m. But if we alreadyhave (n− 1)/2 < r ≤ n− 1, then d = 1 in (7.1.1). So

m(r, n) =n− r

2if

n− 12

< r ≤ n.

In this case any integer m < (n− r)/2 is admissible in Theorem 7.1.Under what restriction on r = dim X can we derive a non-density statement from

Theorem 7.1(i) if the points in question lie in X ∩ C(Hn1 , ε), have bounded height, and

ε > 0 is small? Say n ≥ 4 and 1 ≤ r ≤ n − 3, then one can always take m = 1 inTheorem 7.1. Indeed by Lemma 7.8 we have m(r, n) ≥ m(n− 3, n). If n ≥ 6 thenm(n− 3, n) = 3/2 follows directly from the definition (7.1.1); moreover m(2, 5) = 7/4and m(1, 4) = 17/8. But as m(n− 2, n) = 1 for all n ≥ 4 we cannot apply Theorem 7.1in the critical case r = n− 2 and m = 1.

From the definition of m it is not difficult to deduce

m(r, n) ≤ n− r − 1 if 2 ≤ r ≤ n− 2.

This inequality shows that if 2 ≤ r ≤ n− 2 one can never apply Theorem 7.1 with thecritical subgroup dimension n− r − 1.

We recall that Kronecker’s Theorem implies C(Hnm, 0) = Hn

m; Theorem 7.1(ii) istherefore already known to hold with the best-possible m = n−r−1 and even with Xoa

replaced by Xta if we allow ε = 0, cf. Theorem 6.2. This result follows from the work ofBombieri, Masser, and Zannier. Their approach uses a higher dimension Lehmer-typelower bound for heights relative to the maximal abelian extension of Q due to Amorosoand David. Unfortunately, no way is known to directly reduce Theorem 7.1 to theresult of Bombieri, Masser, and Zannier. The proof of Theorem 7.1 seems to be a newapproach to this sort of problem since a version of this theorem in the setting of abelianvarieties implies new results. The end of the present contains a discussion of this line ofthought. We will deduce Theorem 7.1 using a Bogomolov type lower bound for heightsgiven in the work [AD03] of Amoroso and David.

By Theorem 6.1 we know that Xoa ∩ C(Hnm, ε) has bounded height for small ε if

m < n/r. Hence we have the following corollary:

Corollary 7.1. Let X ⊂ Gnm be an irreducible closed subvariety defined over Q.

7. A BOGOMOLOV PROPERTY MODULO ALGEBRAIC SUBGROUPS 87

(i) If X is a curve then there exists ε > 0 depending only on hV (X), deg(X), andn such that Xoa ∩ C(Hn

n−2, ε) is finite.(ii) If 1 ≤ r = dim X and if m < min{n/r,m(r, n)} is an integer then there exists

ε > 0 such that Xoa ∩ C(Hnm, ε) is finite.

(iii) If dim X ≤ n − 3 and n ≥ 4 then there exists ε > 0 such that Xoa ∩ C(Hn1 , ε)

is finite.

In Corollary 6.2 we showed the finiteness of Xoa∩H52 for a surface X in G5

m. In viewof the fact that the height is bounded on Xoa ∩ C(H5

2, ε) for some ε > 0 by Theorem6.1, it is tempting to try to prove finiteness of this set with a (possibly smaller) positiveε. Unfortunately as m(2, 5) = 7/4 we cannot take m = 2 in Theorem 7.1(ii). Provingthe finiteness of Xoa ∩ C(H5

2, ε) for some positive ε remains an open problem.For curves that are not contained in a proper coset Corollary 7.1(i) is a generalization

of Theorem 2 of Bombieri, Masser, and Zannier in [BMZ99], but also a generalizationof the Bogomolov property. Indeed for n ≥ 2 the set C(Hn

n−2, ε) contains all algebraicpoints of Gn

m of height ≤ ε. (The Bogomolov property holds for the more general classof curves which are not the translate of an algebraic subgroups by a torsion point.)So Corollary 7.1(i) can be viewed as a sort of Bogomolov property for curves modulosubgroups of dimension n − 2. If n = 2 this corollary follows from the Bogomolovproperty since C(H2

0, ε) is precisely the set of points with height ≤ ε; we get nothingnew here.

Let H ⊂ Gnm(Q) be any subset and ε ≥ 0. In chapter 6 we defined the “cone”

C(H, ε) around H; we now define the “tube” around H:

T (H, ε) = {ab; a ∈ H, b ∈ Gnm(Q), h(b) ≤ ε}.

Corollary 7.1(i) motivates the following definition: let n ≥ 2 and let X ⊂ Gnm be an

irreducible algebraic curve, we define

µessC (X) = sup{ε ≥ 0; X ∩ C(Hn

n−2, ε) finite}

where by convention sup ∅ = −∞. We also define

µessT (X) = sup{ε ≥ 0; X ∩ T (Hn

n−2, ε) finite}.

Then clearly µessC (X) ≤ µess

T (X). We are interested in lower bounds for µessC (X) or

µessT (X).

In the special situation n = 2 we have µessT (X) = µess

C (X) = µess(X) where µess(X) isthe usual essential minimum defined in chapter 5. For example in [Zag93] Zagier gavean explicit positive lower bound for µess(X) if X ⊂ G2

m is the line defined by x + y = 1.Although he worked with the slightly different height h(x) + h(y) ≤ 2h(x, y).

Let us study µessC (X) in the so-called geometric case, i.e. if the curve X is not

contained in a proper coset. Corollary 7.1(i) says that µessC (X) can be bounded below

in terms of deg(X), hV (X), and n only. If we consider again the case n = 2, the resultsof Amoroso and David from [AD03] imply that µess

C (X) = µess(X) is bounded belowonly in terms of deg(X) and n. We pose the following question.


Question 1. Let X ⊂ Gnm be an irreducible curve not contained in a proper coset,

is there a positive lower for µessC (X) in terms of deg(X) and n only?

What happens in the arithmetic case, i.e. if we allow the curve X to be containedin a proper coset but not in a proper algebraic subgroup? As was stated in chapter6, X ∩ Hn

n−1 does not necessarily have bounded height. Conjecture A of Bombieri,Masser, and Zannier’s work [BMZ06a] expects that X ∩ Hn

n−2 is finite, or in otherwords µess

C (X) ≥ 0. This conjecture was proved by Maurin in [Mau06]. We consideran example:

Let X ⊂ Gnm be contained in a coset of codimension 2. After an automorphism

of Gnm we may assume X = {(γ1, γ2)} × X ′ where X ′ ⊂ Gn−2

m is a curve. Let ε be apositive real, then any p ∈ X(Q) can be written as p = ab with a = (1, 1, p′) ∈ Hn

n−2 andb = (γ1, γ2, 1, . . . , 1). Hence if h(p) is large with respect to ε and h(b) then h(p′) = h(a)will be large with respect to ε and h(b). Therefore p ∈ C(Hn

n−2, ε) if h(p) is large. SoX ∩ C(Hn

n−2, ε) is infinite for all positive ε. Hence µessC (X) ≤ 0 and so µess

C (X) = 0 byMaurin’s Theorem.

We note that this example does not imply that µessT (X) ≤ 0 if we also assume that

X is not contained in a proper algebraic subgroup. Indeed under this extra hypothesisγ1 and γ2 cannot be roots of unity. By Kronecker’s Theorem our b above satisfiesh(b) = h(γ1, γ2) > 0. We pose a further question.

Question 2. Let X ⊂ Gnm be an irreducible curve not contained in a proper algebraic

subgroup, does µessT (X) > 0 hold?

We use Dobrowolski’s classical theorem to prove a not completely immediate corol-lary to Theorem 7.1:

Corollary 7.2. Let X ⊂ Gnm be an irreducible closed subvariety defined over Q.

(i) If X is a curve then there exists ε > 0 such that Xoa ∩C(Hnn−2, ε) is finite and

equal to Xoa ∩Hnn−2.

(ii) If 1 ≤ r = dim X and if m < min{n/r,m(r, n)} is an integer then there existsε > 0 such that Xoa ∩ C(Hn

m, ε) is finite and equal to Xoa ∩Hn1 .

(iii) If dim X ≤ n − 3 and n ≥ 4 then there exists ε > 0 such that Xoa ∩ C(Hn1 , ε)

is finite and equal to Xoa ∩Hn1 .

So for small ε there are no points on a curve X that are close to a subgroup ofdimension n− 2 without already lying on a subgroup of dimension n− 2 (if Xoa = X).In Corollary 7.2(i) the quantity ε may also depend on the field of definition of X, evenwhen C is replaced by T , as is illustrated by the following simple example.

Let X be the curve defined by x + y = 21/k + 1 with k ∈ N. Then deg(X) = 1 andhV (X) is bounded independently of k. But X contains the non-torsion point (21/k, 1)which lies in T (Hn

0 , ε) with ε = (log 2)/k.We conclude the introduction by discussing the abelian situation. More specifically

we replace Gnm by En where E is an elliptic curve. The set Hn

m also makes sense inthis setting, as do T (·, ·) and C(·, ·) when using for example the Neron-Tate height


associated to an ample symmetric line bundle. Let X ⊂ En be an irreducible curve.Intersections of X with Hn

m have been studied by Viada in [Via03] and Remond andViada in [RV03].

Say X is not contained in the translate of a proper algebraic subgroup of En. If Edoes not have complex multiplication, then Viada proves the finiteness of X ∩ Hn

m form ≤ n/2− 2. If E has complex multiplication, Viada shows that one has finiteness forthe optimal m = n− 2. In her proof she uses a height upper bound analog to Theorem6.1 and also a lower bound for the Neron-Tate height on powers of elliptic curves. Thereason for the apparently non-optimal n/2 − 2 in the non-complex multiplication casecomes from the fact that up to now no one can prove essentially best-possible Lehmer-type height lower bounds here. The problem can be traced back to the fact that on anarbitrary elliptic curves the Frobenius automorphism cannot in general be lifted fromthe reduction modulo a suitable prime.

The work of Amoroso and David in [AD03] on Bogomolov-type lower bounds forheights in the toric case uses methods from transcendence theory. Their extrapolationstep does not make use of the Frobenius automorphism. Thus there is hope that lowerbounds of the same quality as in [AD03] hold in the case where the torus is replacedby a power of an elliptic curve. In fact Galateau has recently announced such a resultin the more general case of a product of elliptic curves in [Gal07].

In a preprint [Via07], Viada improved on her result obtained together with Remond.She proved that if X is a curve not contained in a proper algebraic subgroup, thenX ∩ Hn

n−3 is finite regardless of C.M. type of the elliptic curve E. The subgroup di-mension n− 3 is already close to the best-possible value n− 2. Her approach, obtainedindependently from ours, is based on Galateau’s Bogomolov-type height lower bound.

Let X be a curve in En not contained in the translate of a proper algebraic subgroup.Galateau’s result together with the methods presented in this chapter and the previousone could provide a proof for the finiteness of X∩C(Hn

n−2, ε) for some ε > 0 regardless ifE has complex multiplication or not. Of course such a result would imply the finitenessof X ∩Hn

n−2, which is still unknown.

2. Auxiliary results

We reuse notation introduced in section 2 of chapter 6. For example | · | denotes theeuclidean norm on Rn and also the euclidean norm of the coefficient vector of a linearform. Unless otherwise stated the symbols c1, . . . , c14 denote constants which dependonly on n.

We recall two lemmas from chapter 6 section 3 which will also be used in this chapter.

Lemma 7.1. Let 1 ≤ m ≤ n and let a ∈ Hnm. Then there exist linear forms

L1, . . . , Lm ∈ R[X1, . . . , Xn] such that |Lj | ≤ 1 and

h(au) ≤ c1 max1≤j≤m

{|Lj(u)|}h(a)

for all u ∈ Zn.


The second lemma wraps up all the geometry of numbers we will use.

Lemma 7.2. Let 1 ≤ m ≤ n and let L1, . . . , Lm ∈ R[X1, . . . , Xn] be linear formswith |Lj | ≤ 1. If ρ ≥ 1, there exist λ1, . . . , λn with 0 < λ1 ≤ λ2 ≤ · · · ≤ λn and linearlyindependent u1, . . . , un ∈ Zn such that

|uk| ≤ λk, |Lj(uk)| ≤ ρ−1λk, and λ1 · · ·λn ≤ c2ρm.

We recall Dobrowolski’s Theorem: if α ∈ Q∗ is not a root of unity and D = [Q(α) :Q], then

(7.2.1) h(α) ≥ c31D

(log log 3D

log 2D

)3

where c3 > 0 is an absolute constant.The geometry of numbers machinery and Dobrowolski’s Theorem give the following

lemma which is the main ingredient in the proof of Corollary 7.2. It can be regarded asa Dobrowolski type result modulo subgroups.

Lemma 7.3. Let 1 ≤ m ≤ n be an integer, let δ > 0, and let p ∈ C(Hnm, ε) with

h(p) ≤ B and [Q(p) : Q] ≤ D. If

(7.2.2) ε−1 ≥ c4(1 + B)m+1Dm+1+δ.

then p ∈ Hnm. Here c4 > 0 depends only on n and δ.

Proof. The symbols c′1, c′2, c

′3 denote constants which depend only on n and δ. We

may assume c4 ≥ 2n; we will see how to choose c4 appropriately further down. Wedefine ρ ≥ 1 to be the right-hand side of (7.2.2).

We write p = ab with a ∈ Hnm and h(b) ≤ ε(1 + h(a)). By (6.4.1) we have h(a) ≤

1 + 2B and h(b) ≤ 2ε(1 + B). Let L1, . . . , Lm be the linear forms from Lemma 7.1 andlet λk and uk be from Lemma 7.2 applied to the Lj . We deduce

(7.2.3) h(auk) ≤ c1λkρ−1h(a) ≤ 2c1λkρ

−1(1 + B).

Furthermore, by elementary height inequalities we have h(buk) ≤√

n|uk|h(b) ≤2√

nλkε(1 + B). We combine this inequality with (7.2.3) and use ε ≤ ρ−1 to get

(7.2.4) h(puk) ≤ h(auk) + h(buk) ≤ c′1λk(ρ−1 + ε)(1 + B) ≤ 2c′1λkρ−1(1 + B).

Say 1 ≤ k ≤ n−m. By Lemma 7.2 we have λk ≥ |uk| ≥ 1 and

λk ≤ (λn−m · · ·λn)1

m+1 ≤ (λ1 · · ·λn)1

m+1 ≤ c′2ρm

m+1 .

We apply this inequality to (7.2.4) and use the definition of ρ to get

h(puk) ≤ c′3ρ− 1

m+1 (1 + B) = c′3c− 1

m+1

4 D−1− δm+1 ,

this inequality holds for all 1 ≤ k ≤ n−m.Now [Q(puk) : Q] ≤ D, so if c4 is large enough with respect to c′3 and δ we use

Dobrowolski’s Theorem (7.2.1) to deduce that pu1 , . . . , pun−m are roots of unity. Sinceu1, . . . , un−m are linearly independent we conclude that p ∈ Hn

m by Proposition 6.2. �


Say X ⊂ Gnm is an algebraic curve not contained in a proper coset. Then by Theorem

6.1 chapter 6 there is a B and ε > 0 such that if p ∈ X ∩ C(Hnn−2, ε) then h(p) ≤ B.

By the lemma above we conclude that p is already contained in a algebraic subgroup ofdimension n−2 if ε is small with respect to B, deg(X), [Q(p) : Q], and n. In this case pis contained in the set X ∩Hn

n−2, which is finite by the result of Bombieri, Masser, andZannier mentioned in the introduction. Of course we cannot conclude the finiteness of

{p ∈ X ∩ C(Hnn−2, ε); h(p) ≤ B}

for some fixed positive ε since as of yet we know nothing about the degree [Q(p) : Q].Also, if we already knew that the degrees [Q(p) : Q] were uniformly bounded, thenfiniteness would follow from boundedness of height and Northcott’s Theorem.

3. Push-forwards and pull-backs

In this section we prove two lemmas on bounds for degrees of push-forwards andpull-backs of varieties by a homomorphism of algebraic groups.

Let X ⊂ Gnm be an irreducible subvariety throughout this section.

Lemma 7.4. Let u1, . . . , ut ∈ Zn with 0 < |u1| ≤ · · · ≤ |ut| and ϕ = ϕ(u1,...,ut) :Gn

m → Gtm. If q = dim ϕ(X), then

deg(ϕ(X)) ≤ c5|ut−q+1| · · · |ut|deg(X).

Proof. For brevity we set Y = ϕ(X). By Theorem 5.3 there exists U ⊂ YZariski open and dense with U ⊂ ϕ(X). By a standard Bertini type argument wemay find polynomials li ∈ Q[X1, . . . , Xt] (1 ≤ i ≤ q) of degree 1 such that li−Xt−q+i ∈Q[X1, . . . , Xt−q] with the following properties: the set

S = {y ∈ Y ; l1(y) = · · · = lq(y) = 0}

is finite of cardinality deg(Y ) and contained in U . We define R = ϕ|−1X (S). Then

R ⊂ X is Zariski closed and has at least deg(Y ) irreducible components. On the otherhand we have R = {x ∈ X; l1(ϕ(x)) = · · · = lq(ϕ(x)) = 0, }. The exponent vectorsin li ◦ ϕ have norm bounded by |ut−q+i|. Hence Bezout’s Theorem 5.4 implies that thenumber of irreducible components of R is bounded above by c5|ut−q+1| · · · |ut|deg(X).This completes the proof. �

If ϕ = π : Gnm → Gt

m is the projection onto any set of t coordinates, then proofabove can easily be modified to give deg(π(X)) ≤ deg(X).

Lemma 7.5. Let u1, . . . , un ∈ Zn be linearly independent and ϕ = ϕ(u1,...,un) : Gnm →

Gnm. Then there exists an irreducible component W ⊂ ϕ−1(X) such that ϕ(W ) ⊂ X is

Zariski dense,

dim W = dim X, and deg(W ) ≤ c6|u1| · · · |un|deg(X).


Proof. Let W1, . . . ,Wl be the irreducible components of ϕ−1(X). Since ϕ is sur-jective ϕ(Wi0) ⊂ X is Zariski dense for some i0. We set W = Wi0 . Because ϕ has finitefibres we conclude that dim W = dim X by Theorem 5.1.

It remains to prove the upper bound for deg(W ). By Bezout’s Theorem (Theorem5.4) the irreducible components of

{(p, q) ∈ Gnm ×Gn

m; ϕ(p) = q} ∩ (Gnm ×X)

have degree bounded by c6|u1| · · · |un|deg(X). Now W is the projection of such anirreducible component onto the first factor of Gn

m ×Gnm. The lemma follows from the

remark after the proof of Lemma 7.4. �

4. A lower bound for the product of heights

In [AD03] Amoroso and David deduced a positive lower bound for the height of apoint on an open subset of certain subvarieties of Gn

m, cf. Theorem 5.7. Their bounddepends only on the degree of the variety and on n. In this section we derive a corollaryof this result by deducing a lower bound for the product over heights of some coordinatesof a generic point on a variety. The lower bounds depends only on the degree of thevariety and n.

Let us consider for the moment the curve X in G3m defined by p1 + p2 = 1 and

p2 + p3 = 2 (we have Xoa = X). Because of examples like (21/k, 1− 21/k, 1 + 21/k) ∈ Xwith k ∈ N the product

(7.4.1) h(p1)h(p2)h(p3)

cannot be bounded below by some positive constant on any Zariski open dense subsetof X. The trick is to forget about the coordinate in (7.4.1) with minimal height, in thiscase p1 if k is large. In Proposition 7.1 below we will show that for all p on a open densesubset X the quantity

max{h(p1)h(p2), h(p1)h(p3), h(p2)h(p3)}

is bounded below by a positive constant.First we need a preparatory lemma which encapsulates the consequence (5.3.5) of

Amoroso and David’s Theorem (Theorem 5.7).

Lemma 7.6. Let X ( Gnm be an irreducible closed subvariety that is not contained in

a proper coset. Let u1, . . . , un ∈ Zn be linearly independent and ϕ = ϕ(u1,...,un) : Gnm →

Gnm. We set Π = |u1| · · · |un|. There exists a Zariski open dense U ⊂ X such that if

q ∈ Gnm with ϕ(q) ∈ U , then

h(q) ≥ c7

(Π deg(X))1/ codim X(log(3Π deg(X)))c8

where c7 and c8 are positive and depend only on n.


Proof. Let W ⊂ ϕ−1(X) be an irreducible component as in Lemma 7.5. ClearlyW is not contained in a proper coset of Gn

m. By (5.3.5), the consequence of Amorosoand David’s Theorem, we have

µess(W ) ≥ c

(Π deg(X))1/ codim X(log(3Π deg(X)))c′> 0,

where c and c′ depend only on n. By definition of the essential minimum there existsV ⊂ W Zariski open and dense such that if q ∈ V (Q) then h(q) ≥ 1

2 µess(W ). Finallyby Theorem 5.3 there exists U ⊂ ϕ(V ) such that U ⊂ X is Zariski open and dense. Thelemma follows because if ϕ(q) ∈ U then q = q′ζ for some q′ ∈ V and some torsion pointζ; thus h(q) = h(q′). �

The following proposition is a geometric analogue of Theorem 1.6 in [AD99].

Proposition 7.1. Let X ⊂ Gnm be an irreducible closed subvariety that is not con-

tained in a proper coset. There exists U ⊂ X Zariski open and dense such that for eachp = (p1, . . . , pn) ∈ U(Q) there is a subset Σ ⊂ {1, . . . , n} with |Σ| ≥ n− dim X and

(7.4.2)∏k∈Σ

h(pk) ≥c9

deg(X)(log(3 deg(X)))c10

with c9 > 0.

Proof. Throughout the proof c′1, . . . , c′6 will denote constants which depend only

on n. For brevity let r = dim X. Clearly we may assume that X 6= Gnm. We prove the

proposition by contradiction. To do this we assume that there exists V ⊂ X(Q) Zariskidense such that for all p = (p1, . . . , pn) ∈ V and all Σ ⊂ {1, . . . , n} with |Σ| ≥ n− r wehave

(7.4.3)∏j∈Σ

h(pj) <C

deg(X)(log(3 deg(X)))C′

where C,C ′ > 0 are fixed and depend only on n. We will see how to choose theseconstants properly later on.

First we do some reduction steps. By replacing V by a smaller, yet still Zariskidense subset of X(Q) and after permuting coordinates we may assume that

h(p1) ≤ h(p2) ≤ · · · ≤ h(pn)

for all (p1, . . . , pn) ∈ V . These inequalities will be used freely throughout the proof.Let us temporarily assume that the set of (p1, . . . , pn) ∈ V with h(pn) > 1 is dense

in X. Let X ′ ⊂ Gn−1m be the Zariski closure of the projection of X onto the first n− 1

coordinates. Now dim X ′ ≤ r and X ′ has degree at most deg(X) by the remark afterLemma 7.4. Also, X ′ is not contained in a proper coset. By induction on n and using(7.4.3) with Σ = {r + 1, . . . , n} we conclude a contradiction if C is small enough and C ′

is big enough. Hence we may suppose that

(7.4.4) maxj{h(pj)} = h(pn) ≤ 1 for (p1, . . . , pn) ∈ V,


where V is still Zariski dense in X.Let p = (p1, . . . , pn) ∈ V be arbitrary for the moment. Let π : Gn

m → Gr+1m be the

projection onto the first r + 1 coordinates. The variety Y = π(X) is not contained ina proper coset since X itself is not. We note that deg(Y ) ≤ deg(X) by the commentafter Lemma 7.4 and codim Y ≥ 1. Thus (5.3.5) gives a lower bound for µess(Y ). Soafter replacing V by a smaller, dense subset of X(Q) we may assume that

(r + 1)h(pr+1) = (r + 1) max{h(p1), . . . , h(pr+1)} ≥ h(π(p)) ≥(7.4.5)

≥ c′1deg(X)(log(3 deg(X)))c′2

.

In particular we have h(pr+1) > 0. For r + 1 ≤ j ≤ n we define

(7.4.6) k1 = k2 = · · · = kr = 1 and kj =[

h(pj)h(pr+1)

]≥ 1.

We note kr+1 = 1. The kj depend on the point p. For r + 1 ≤ j ≤ n we have

(7.4.7)12

h(pj)h(pr+1)

≤ kj ≤h(pj)

h(pr+1)

Next we would like to bound the kj from above. For r + 1 ≤ j ≤ n we apply (7.4.4)and (7.4.5) to deduce

(7.4.8) 1 ≤ kj ≤1

h(pr+1)≤ c′3 deg(X)(log(3 deg(X)))c′2 ,

with c′3 ≥ 1. In fact (7.4.8) holds for all 1 ≤ j ≤ n. As an important consequence wesee that the kj are bounded independently of p ∈ V .

Still considering p ∈ V we define a homomorphism ϕ : Gnm → Gn

m by ϕ(x1, . . . , xn) =(xk1

1 , . . . , xknn ) where the kj are as in (7.4.6). Of course ϕ depends on p. Each such ϕ

gives rise to an open dense Uϕ ⊂ X as in Lemma 7.6. Since the set of possible ϕ, i.e.with kj satisfying (7.4.8), is finite and independent of p there are only finitely many Uϕ.Because V ⊂ X is Zariski dense we may pick once and for all a p ∈ V with p ∈ Uϕ forall ϕ as defined above with kj satisfying (7.4.8).

Let q = (q1, . . . , qn) ∈ Gnm(Q) with ϕ(q) = p. By our choice of p and Lemma 7.6 we

have the lower bound

(7.4.9) h(q) ≥ c7

(k1 · · · kn deg(X))1/(n−r)(log(3k1 · · · kn deg(X)))c8.

We have

(7.4.10) h(pr+1) · · ·h(pn) = kr+1 · · · knh(qr+1) · · ·h(qn).

From (7.4.7) we derive

(7.4.11) 1 ≤ h(qj)h(pr+1)

≤ 2 for r + 1 ≤ j ≤ n.


We apply the lower bound in (7.4.11) to the right side of (7.4.10) and get

(7.4.12) h(pr+1) · · ·h(pn) ≥ kr+1 · · · knh(pr+1)n−r.

Say 1 ≤ j′ ≤ n with h(qj′) = max{h(q1), . . . , h(qn)}. Now h(qr+1) = h(pr+1) ≥h(pj) = h(qj) for 1 ≤ j ≤ r, hence we may assume j′ ≥ r + 1. We insert the upperbound from (7.4.11) with j = j′ into (7.4.12) and use h(qj′) ≥ 1

nh(q) to derive

h(pr+1) · · ·h(pn) ≥ 2−(n−r)kr+1 · · · knh(qj′)n−r ≥ c′4kr+1 · · · knh(q)n−r.

We use the lower bound for h(q) in (7.4.9) to get

h(pr+1) · · ·h(pn) ≥ c′5kr+1 · · · kn

k1 · · · kn deg(X)(log(3k1 · · · kn deg(X)))(n−r)c8

=c′5

deg(X)(log(3k1 · · · kn deg(X)))(n−r)c8.

We bound the remaining kj in the logarithm from above with the help of (7.4.8) andobtain

(7.4.13) h(pr+1) · · ·h(pn) ≥ c′6deg(X)(log(3 deg(X)))(n−r)c8

.

If we choose C = c′6 and C ′ = (n− r)c8, then (7.4.3) and (7.4.13) contradict. �

It is not too difficult to construct examples of X of arbitrary dimension, as atthe beginning of this section, where one really needs to omit dim X factors in theproduct (7.4.2) to obtain a positive lower bound on an open dense set. Furthermore,the dependency of (7.4.2) on deg(X) is optimal up to the usual log power.

We will prove a variant of Proposition 7.1 where the variety is allowed to be containedin a proper coset. Let X ⊂ Gn

m, we define

so(X) = inf{dim H; H ⊂ Gnm a coset with X ⊂ H}.

Proposition 7.1 only holds for varieties X ⊂ Gnm with so(X) = n. A simple projec-

tion argument shows that in general we have:

Proposition 7.2. Let X ⊂ Gnm be an irreducible closed subvariety. There exists

U ⊂ X Zariski open and dense such that for each p = (p1, . . . , pn) ∈ U(Q) there is asubset Σ ⊂ {1, . . . , n} with |Σ| ≥ so(X)− dim X and∏

k∈Σ

h(pk) ≥c11

deg(X)(log(3 deg(X)))c12

with c11 > 0.

Proof. Let H be a coset with X ⊂ H and so(X) = dim H = n−h. By Proposition6.2 there are linearly independent u1, . . . , uh ∈ Zn such that xui is constant on H.We may assume that the h × h matrix whose ith row consists of the first h entries ofui is non-singular. In this case the projection π : Gn

m → Gn−hm onto the last n − h

coordinates has finite fibres when restricted to H. Therefore π|X has finite fibres too.


By Theorem 5.1 we have dim π(X) = dim X. The comment after Lemma 7.4 impliesdeg(π(X)) ≤ deg(X). Furthermore, π(X) is not contained in a proper coset, indeedotherwise X would be contained in a coset of dimension strictly less than n − h. Theproposition now follows from Proposition 7.1 applied to π(X). �

If for example X is itself a coset, then so(X) = dim X; in this case Proposition 7.2is an empty statement.

5. Proof of Theorem 7.1 and corollaries

We start off with an auxiliary lemma on linear programming. We recall that m(r, s)was defined in (7.1.1).

Lemma 7.7. Let 1 ≤ r < s ≤ n be integers, let M = (mij) be the n× (s− r) matrixdefined by

mij =

1 if i + j ≤ n− r + 1,2 if n− r + 2 ≤ i + j ≤ n + 1,0 else,

and let w = (w1, . . . , ws−r)t ∈ Rs−r be the column vector with

wj ={

2−( j−1r

+1) if r|(j − 1),0 else.

Then v = (v1, . . . , vn)t = Mw satisfies vi ≤ 1 and furthermore

(7.5.1)s−r∑j=1

(s− r − j + 1)wj = m(r, s).

Proof. The vector w looks like

(7.5.2) w = (12, 0, . . . , 0,

14, 0, . . . , 0,

18, 0, . . . )

t

with r − 1 zeros between consecutive negative powers of 2 (there are no zeros if r = 1).When 1 ≤ i ≤ n, the ith row of M starts off with a certain number (possibly zero) ofconsecutive ones followed by say N consecutive twos and finally consecutive zeros. Bydefinition we have N ≤ r, hence by (7.5.2) there is at most one j with mij = 2 andwj 6= 0. Let N ′ be the number of j with mij = 1 and wj 6= 0, then

vi =∑

j, mij=1

wj + 2∑

j, mij=2

wj ≤ (12

+ · · ·+ 12N ′ ) + 2

12N ′+1

= 1.

This inequality proves the first part of the lemma. The second part, the equality (7.5.1),follows from an elementary calculation. �

The following proposition will imply part (i) of Theorem 7.1.


Proposition 7.3. Let X ⊂ Gnm be an irreducible closed subvariety of dimension

r ≥ 1 and assume that s = so(X) ≥ r + 1. Let m be an integer with 0 ≤ m < m(r, s),let B ≥ 1, and let δ > 0. Then there exists c13 > 0 which depends only on n and δ suchthat if

(7.5.3) ε ≤ c13(Bm+δ(deg(X))1+δ)−1

m(r,s)−m

the set {p ∈ X ∩ T (Hnm, ε); h(p) ≤ B} is not Zariski dense in X.

Proof. Throughout the proof c′1, . . . , c′12 will denote constants which only depend

on n and δ. We consider c13 ≤ 1 as fixed and depending only on n and δ; we will see howto choose it later on. We also define ρ ∈ R such that Bρ−1 is equal to the right-handside of (7.5.3). We note that ρ ≥ B ≥ 1.

Now say p ∈ X ∩ T (Hnm, ε) and h(p) ≤ B. We will show that p is contained in some

proper Zariski closed subset of X independent of p.By definition we may write p = ab with a ∈ Hn

m and h(b) ≤ ε. Elementary heightproperties imply h(a) ≤ h(p) + h(b−1) ≤ h(p) + nh(b) ≤ h(p) + nε ≤ 2nB. Let usassume for the moment that m ≥ 1. By Lemmas 7.1 and 7.2 there exist λ1, . . . , λn with0 < λ1 ≤ · · · ≤ λn and linearly independent u1, . . . , un ∈ Zn such that for 1 ≤ k ≤ n

(7.5.4) |uk| ≤ λk, h(auk) ≤ c′1h(a)ρ−1λk ≤ c′2Bρ−1λk, and λ1 · · ·λn ≤ c′3ρm.

In the case m = 0 the statements in (7.5.4) also hold if we take λk = 1 and uk thestandard basis elements of Rn. Indeed if m = 0, then a is a torsion point and thus hasheight 0. Elementary height inequalities give h(buk) ≤

√nh(b)|uk| ≤

√nελk, hence

(7.5.5) h(puk) ≤ h(auk) + h(buk) ≤ c′4(Bρ−1 + ε)λk ≤ 2c′4Bρ−1λk,

here we used the bound ε ≤ Bρ−1 in the last inequality.Let t0 = n−s+r+1, so that r+1 ≤ t0 ≤ n; also let t be an integer with t0 ≤ t ≤ n.

We define the surjective homomorphism of algebraic groups ϕt = ϕ(u1,...,ut) : Gnm → Gt

m

and the irreducible variety Xt = ϕt(X) ⊂ Gtm. Both ϕt and variety Xt depend on p.

On the other hand the quantity c′3ρm depends only on X, B, and δ but not on the point

p. Since |u1| · · · |ut| ≤ c′3ρm by (7.5.4) there are only finitely many possibilities for ϕt.

These morphisms give a finite set of Xt which does not depend on p.We proceed by bounding so(Xt) from below. To do this let H ⊂ Gt

m be a coset ofdimension so(Xt) that contains Xt. Then dim ϕ−1

t (H) = dim H + (n − t) and ϕ−1t (H)

is a coset containing X. Hence

(7.5.6) so(Xt) = dim ϕ−1t (H)− (n− t) ≥ s + t− n ≥ s + t0 − n = r + 1.

In particular Xt has positive dimension.Lemma 7.4 and (7.5.4) give the upper bound

deg(Xt) ≤ c′5λt−dim Xt+1 · · ·λt deg(X)

since the λk are in ascending order. Furthermore, dim Xt ≤ r and λk ≥ 1, so

(7.5.7) deg(Xt) ≤ c′5λt−r+1 · · ·λt deg(X) for all t0 ≤ t ≤ n.


Let c11 > 0 and c12 be the constants from Proposition 7.2. Let us assume for themoment that there exists an integer t with t0 ≤ t ≤ n such that for all Σ ⊂ {1, . . . , t}with |Σ| ≥ so(Xt)− dim Xt we have the inequality

(7.5.8) c−111

(∏k∈Σ

h(puk)

)deg(Xt)(log(3 deg(Xt)))c12 < 1.

The product in (7.5.8) is actually a product over heights of certain coordinates of thepoint ϕt(p) ∈ Xt. From (7.5.8) and Proposition 7.2 we conclude that ϕt(p) is containedin Zϕ,t, a proper and Zariski closed subset of Xt. As mentioned above, the set of possibleϕt and Xt is finite. Finally ϕt|X : X → Xt is by definition a dominant morphism. Thepoint p is therefore contained in the proper, Zariski closed subset

⋃ϕ,t ϕt|−1

X (Zϕ,t), wherethe union is taken over a finite set independent of p. The proposition follows in thiscase.

What if p does not satisfy the property described around (7.5.8)? Then we willderive a contradiction. Let t0 ≤ t ≤ n and let Σ ⊂ {1, . . . , t}, we define fn−t+1(Σ) ∈ Rto be the expression on the left-hand side of (7.5.8). We are assuming that for allt0 ≤ t ≤ n there exists a subset Σ(t) ⊂ {1, . . . , t} with |Σ(t)| ≥ so(Xt)− dim Xt and

(7.5.9) fn−t+1(Σ(t)) ≥ 1.

For brevity we set fn−t+1 = fn−t+1(Σ(t)).We continue by bounding fn−t+1 from above. To do this we apply (7.5.5) to the

definition of fn−t+1 and get

fn−t+1 ≤ c−111

∏k∈Σ(t)

(c′6Bρ−1λk)

deg(Xt)(log(3 deg(Xt)))c12 .

Next we bound deg(Xt) for above using (7.5.7):

fn−t+1 ≤ c′7

∏k∈Σ(t)

λk

λt−r+1 · · ·λt(Bρ−1)|Σ(t)| deg(X)(log(3λt deg(X)))c12 .

By (7.5.6) we have |Σ(t)| ≥ so(Xt)−dim Xt ≥ s+t−n−r ≥ 1 where we used dim Xt ≤ r.Since λk ≥ 1, ρ ≥ B, and

∏k∈Σ(t) λk ≤ λ1 · · ·λt we deduce

fn−t+1 ≤(7.5.10)

c′7λ1 · · ·λt−r(λt−r+1 · · ·λt)2(Bρ−1)s+t−n−r deg(X)(log(3λt deg(X)))c12 .

And this inequality holds for all t0 ≤ t ≤ n.Let M , v, w be the matrix respectively vectors from Lemma 7.7. Using notation

from chapter 6, section 2 we define

Λ = (λ1, . . . , λn)M

= (λ1 · · ·λn−r(λn−r+1 · · ·λn)2, . . . , λ1 · · ·λn−s+1(λn−s+2 · · ·λn+r−s+1)2).


That is, the jth entry of Λ is the main contribution of the λk’s to the bound for fj in(7.5.10). By Lemma 7.7 and λk ≥ 1 we have

(7.5.11) Λw = (λ1, . . . , λn)Mw = λv11 · · ·λvn

n ≤ λ1 · · ·λn.

We define the product

f = (f1, . . . , fs−r)w = fw11 · · · fws−r

s−r .

By (7.5.9) and since wj ≥ 0 we conclude

(7.5.12) f ≥ 1.

We bound f from above with the help of (7.5.10) and (7.5.11)

f ≤ c′8Λw(Bρ−1)

Ps−rj=1(s−r−j+1)wj (deg(X)(log(3λn deg(X)))c12)w1+···+ws−r

≤ c′8λ1 · · ·λn(Bρ−1)Ps−r

j=1(s−r−j+1)wj deg(X)(log(3λn deg(X)))c12 .

We also used w1 + · · ·+ ws−r < 1, which follows easily from the definition of the wj inLemma 7.7, cf. (7.5.2). By the same lemma the exponent of Bρ−1 equals m(r, s). Werecall (7.5.4) and observe λn ≤ λ1 · · ·λn ≤ c′3ρ

m to bound

f ≤ c′9ρm−m(r,s)Bm(r,s) deg(X)(log(3ρ deg(X)))c12 .(7.5.13)

By the definition of ρ we have

(7.5.14) ρm−m(r,s)Bm(r,s)+δ deg(X)1+δ = cm(r,s)−m13

Together with (7.5.13) we get

f ≤ c′9cm(r,s)−m13

(log(3ρ deg(X)))c12

Bδ deg(X)δ.

By (7.5.14) the ρ in the logarithm is bounded above by (Bc−113 deg(X))c′10 . Together

with elementary inequalities we get

f ≤ c′11cm(r,s)−m13

(log(3Bc−113 deg(X)))c12

Bδ deg(X)δ(7.5.15)

≤ c′12cm(r,s)−m13

(Bc−113 deg(X))min{m(r,s)−m

2,δ}

Bδ deg(X)δ

≤ c′12cm(r,s)−m

213 .

Thus we may choose c13 ∈ (0, 1] depending only on n and δ such that (7.5.15) impliesf < 1. But this inequality contradicts (7.5.12). �

For B ≥ 1 we have the following inclusions

{p ∈ T (Hnm, ε); h(p) ≤ B} ⊂ {p ∈ C(Hn

m, ε); h(p) ≤ B}(7.5.16)

⊂ {p ∈ T (Hnm, 4εB); h(p) ≤ B},


if ε ≤ 12n . The first inclusion is trivial and holds for unrestricted ε, the second one

follows easily from (6.4.1). Therefore Proposition 7.3 can be reformulated with T (·, ·)replaced by C(·, ·) and after choosing a possibly smaller ε.

Proposition 7.4. Let X, r, s, m, n,B, and δ be as in Proposition 7.3. Then thereexists c14 > 0 which depends only on n and δ such that if

ε ≤ c14(Bm(r,s)+δ(deg(X))1+δ)−1

m(r,s)−m

the set {p ∈ X ∩ C(Hnm, ε); h(p) ≤ B} is not Zariski dense in X.

Proof. The proof follows immediately from the second inclusion in (7.5.16) andProposition 7.3. �

Lemma 7.8. Let n, n′, r, r′ be integers with 1 ≤ r ≤ n − 1, 1 ≤ r′ ≤ n′ − 1, r′ ≤ r,and n− r ≤ n′ − r′, then m(r, n) ≤ m(r′, n′).

Proof. From (7.5.1) and taking j = kr + 1 we get

m(r, n) =∞∑

k=0

max{0,n− (k + 1)r

2k+1}.

Asn− (k + 1)r = (n− r)− kr ≤ (n′ − r′)− kr′ = n′ − (k + 1)r′

for non-negative k, the result follows at once. �

Lemma 7.9. Let X ( Gnm be an irreducible closed subvariety of dimension r ≥ 1.

Let m be an integer with 0 ≤ m < m(r, n) and let B ≥ 1. Say Z ⊂ X is a closedirreducible subvariety of positive dimension with Z ∩Xoa 6= ∅. Then there exists ε > 0such that

{p ∈ Z ∩ C(Hnm, ε); h(p) ≤ B}

is not Zariski dense in Z.

Proof. The hypothesis Z ∩Xoa 6= ∅ implies that Z is not an anomalous subvarietyof X. Say H ⊂ Gn

m is a coset containing Z with so(Z) = dim H. By hypothesis we havedim Z ≤ r + dim H − n or

(7.5.17) so(Z)− dim Z ≥ n− r

and in particular so(Z) ≥ dim Z + 1 since X is proper. Inequalities (7.5.17), dim Z ≤dim X, and Lemma 7.8 imply m(dim Z, so(Z)) ≥ m(r, n) > m. Therefore the lemmafollows from Proposition 7.4 applied to Z. �

Before proving Theorem 7.1 we prove Corollaries 7.1 and 7.2.We start with Corollary 7.1(i). Let X be as in the hypothesis of part (i). We may

assume Xoa = X, i.e. X is not contained in a proper coset. By Theorem 6.1 withs = dim X = 1 the set X ∩C(Hn

n−1, ε) has height bounded by B for an ε > 0 such thatB depends only on hV (X), deg(X), and n. Moreover, ε depends only on deg(X) andn. Since m(1, n) > n− 2 we conclude finiteness of X ∩C(Hn

n−2, ε) from Theorem 7.1(i)


after choosing a possibly smaller ε > 0 which now depends only on hV (X), deg(X), andn.

The proof of part (ii) is similar.For part (iii) we may assume r = dim X ≥ 1. In section 1 we saw that 1 ≤ r ≤ n−3

implies m(r, n) > 1. Hence the proof follows from taking m = 1 in part (ii). �

To prove Corollary 7.2(i) we note that by Corollary 7.1(i) we may choose ε > 0 suchthat Xoa ∩ C(Hn

n−2, ε) is finite. Hence the points of this set have height bounded bysome fixed B and degree bounded by some fixed D. The proof now follows easily fromLemma 7.3 after adjusting ε if necessary.

The proofs of parts (ii) and (iii) are similar. �

Proof of Theorem 7.1: We may assume m ≥ 0, thus X 6= Gnm since m(n, n) = 0. Part

(i) follows from Proposition 7.4 and so(X) = n, so it remains to prove part (ii). We willprove the following statement:

Let Z ⊂ X be an irreducible closed subvariety, then there exists ε > 0 such that

(7.5.18) {p ∈ Xoa ∩ Z ∩ C(Hnm, ε); h(p) ≤ B}

is finite.Of course the theorem follows by taking Z = X in the statement above. We prove

the statement by induction on dim Z. The case dim Z = 0 being trivial we assumedim Z ≥ 1 and also Xoa ∩ Z 6= ∅. By Lemma 7.9 there exists an ε > 0 such that

(7.5.19) {p ∈ Xoa ∩ Z ∩ C(Hnm, ε); h(p) ≤ B} ⊂ Y1 ∪ · · · ∪ Yl

where the Yi ( Z are proper, Zariski closed, and irreducible. As dim Yi ≤ dim Z − 1we reduce ε if necessary and apply the induction hypothesis to conclude that {p ∈Xoa ∩ Yi ∩C(Hn

m, ε); h(p) ≤ B} is finite for each i. The statement around (7.5.18) nowfollows from (7.5.19). �

APPENDIX A

Quasi-equivalence of heights

Say two algebraic number x and y satisfy a simple equation such as xp +yq = 0 withp and q positive integers. Then the elementary height properties described in chapter 1imply that the heights are related as ph(x) = qh(y). Now say P (x, y) = 0 where P isan irreducible polynomial in two variables with algebraic coefficients and p = degX P ,q = degY P , then how are h(x) and h(y) related? In this generality the expression|ph(x) − qh(y)| need not be bounded independently of x and y. Nevertheless in thisappendix we prove an upper bound in terms of max{h(x), h(y)}, hp(P ), p, and q.

1. Introduction

Let C be a smooth projective curve defined over Q. We assume that C is embeddedinto projective space Pn. Let f ∈ Q(C) be a non-constant rational function on C. Weconsider f as a rational map C → A1. Then f has a well-defined degree deg(f), thecardinality of f−1(q) for generic q. Let g ∈ Q(C) be a further non-constant rationalfunction. We would like to compare the heights h(f(p)) and h(g(p)) as p runs over thealgebraic points of C that are not poles of f or g. It was known to Siegel that thequantities h(f(p))/ deg(f) and h(g(p))/ deg(g) are asymptotically equal. One can evenshow

(A.1.1)∣∣∣∣h(f(p))deg(f)

− h(g(p))deg(g)

∣∣∣∣ = O(max{1, h(f(p)), h(g(p))}1/2).

The implied constant may depend on C, f , and g. The inequality (A.1.1) can forexample be proved using the Neron-Tate height on the Jacobian of C if our curve haspositive genus.

As f and g are rational functions on the curve C they are contained in a field oftranscendence degree 1 over Q. Hence f and g satisfy some algebraic relation. That is,there exists a polynomial P ∈ Q[X, Y ] with P (f, g) = 0 on C. We may assume that Pis irreducible over Q and in this case we have

degX P

degY P=

deg g

deg f.

Hence inequality (A.1.1) follows from the following statement:Let P ∈ Q[X, Y ] be irreducible over Q with p = degX P > 0 and q = degY P > 0.

There exists a constant c1 = c1(P ) such that if x and y are algebraic numbers with

103


P (x, y) = 0, then

(A.1.2)∣∣∣∣h(x)

q− h(y)

p

∣∣∣∣ ≤ c1 max{1, h(x), h(y)}1/2.

The purpose of this appendix is to give a proof of (A.1.2) and to determine a constantc1 which is completely explicit in terms of P . We will strive for a good dependency inthe height of P and the partial degrees p and q. For example if we assume that hp(P )is large compared with max{p, q}, then we will show that c1 depends only on hp(P ).

Property (A.1.2) is often referred to as quasi-equivalence of heights. We will nowstate the Quasi-equivalence Theorem:

Theorem A.1. Let P ∈ Q[X, Y ] be irreducible with p = degX P > 0 and q =degY P > 0. If P (x, y) = 0 with x, y ∈ Q, then∣∣∣∣h(x)

q− h(y)

p

∣∣∣∣ ≤ 51 max{p, q, hp(P )}1/2 max{1, h(x), h(y)}1/2.

In [Abo06] Abouzaid proved a variant of the Quasi-equivalence Theorem. The de-pendency in hp(P ) and max{h(x), h(y)} is essentially the same in both version. Thoughin Abouzaid’s bound, the dependence on the partial degrees is slightly worse.

It is essential that P is irreducible. For example the partial degrees of (X2−Y )(X−Y 2) are equal to 3, but clearly (A.1.2) cannot hold for this polynomial. Although it ispossible to formulate a version of Theorem A.1 with P ∈ K[X, Y ] where K is a numberfield and P is irreducible over K. In this case P is up to a scalar factor the product ofpolynomials which are irreducible in Q[X, Y ] and conjugated over K. Thus all factorshave equal partial degrees.

Sometimes it is useful to bound h(y) uniformly in terms of h(x) if the height of x islarge:

Corollary A.1. Let P be as in Theorem A.1. Say P (x, y) = 0 with x, y ∈ Q and

(A.1.3) h(x) ≥ 2.104(deg P )2 max{p, q, hp(P )},then h(y) ≤ 2p

q h(x).

The proof of Theorem A.1 depends on the Absolute Siegel Lemma by Roy andThunder and some elementary theory of algebraic functions. We note that a proof alongthe lines of the proof of Theorem A.1 is possible using Bombieri and Vaaler’s classicalversion of Siegel’s Lemma. But this comes at the cost of introducing a dependency onthe field of definition of P , more precisely the discriminant of this field, in c1 of (A.1.2).

We give a short sketch of the proof of Theorem A.1. Let m and n be large integerswith n/m approximately equal to p/q. In section 2 we use the Absolute Siegel Lemma toconstruct polynomials A and B in two variables with algebraic coefficients of boundedheight such that P divides AY m − B as a polynomial. Furthermore, we require thatdegX A,degX B ≤ n and degY A,degY B < degY P . If P (x, y) = 0, then A(x, y)ym =B(x, y). If we assume for the moment A(x, y) 6= 0, then we may bound the height of yin terms of the height of x by using the product formula and the fact that m is much

A. QUASI-EQUIVALENCE OF HEIGHTS 105

larger than degY P . In section 3 we show that if A(x, y) = 0 then the order of vanishingcannot be too large if m and n are well chosen. We then differentiate appropriatelyand replace A,B by new polynomials A′, B′ with controlled height and degree such thatA′(x, y)ym = B′(x, y) and A′(x, y) 6= 0. Thus again we get a height bound of y. Byinterchanging x and y we can bound the height of x by the height of y and this completesthe proof. We note that the proof does not depend on an explicit version of Eisenstein’sTheorem.

2. Construction via Siegel’s Lemma

If K is a number field with an absolute value | · | = | · |v (v ∈ MK) and P is apolynomial with coefficients in K in any number of variables, then, just as in chapter 1,we define |P | to be the maximum of the absolute values of the coefficients of P . We alsorecall that the height hp(P ) was defined in chapter 1. If Q is a further polynomial withalgebraic coefficients it will sometimes be useful to define hp(P,Q) as the projectiveheight of the vector consisting of the coefficients of P and Q. Finally δv and dv aredefined in chapter 1.

We start off by proving a standard lemma concerning simple properties of absolutevalues and heights.

Lemma A.1. Let K be a field and A,B ∈ K[X, Y ].(i) If K is a number field and v ∈ MK , then

|A + B|v ≤ δv(2)max{|A|v, |B|v},|AB|v ≤ δv(1 + min{degX A, degY A,degX B,degY B})|A|v|B|v.

(ii) If K = Q and x, y ∈ Q with A(x, y) 6= 0, then

h(B(x, y)/A(x, y)) ≤ hp(A,B) + max{degX A,degX B}h(x)

+ max{degY A,degY B}h(y)

+ log max{(1 + degX A)(1 + degY A), (1 + degX B)(1 + degY B)}

(iii) If K = Q, x, y ∈ Q with A(x, y) = 0 and assume furthermore that A is notdivisible in Q[X, Y ] by X − α for some α ∈ Q, then

(A.2.1) h(y) ≤ (degX A)h(x) + log((1 + degX A) degY A) + hp(A).

Proof. The first inequality in (i) follows directly from the triangle inequality. Toprove the second inequality we write A =

∑i,j aijX

iY j and B =∑

i,j bijXiY j . Then

AB =∑

i,j cijXiY j with

cij =∑

i′+i′′=i,j′+j′′=j

ai′j′bi′′j′′ .

The sum above involves at most 1 + min{degX A,degY A, degX B,degY B} non-zeroterms. Hence the desired inequality follows from the triangle inequality.


Now to part (ii). We note that the product formula (1.1.1) implies

(A.2.2) h(B(x, y)/A(x, y)) = [F : Q]−1∑

v∈MF

dv log max{|A(x, y)|v, |B(x, y)|v}

where F is a number field containing x, y and the coefficients of A and B. Note thatthe polynomial A involves at most (1+degX A)(1+degY A) non-zero coefficients, hencethe triangle inequality gives

|A(x, y)|v ≤

δv((1 + degX A)(1 + degY A))|A|v max{1, |x|v}degX A max{1, |y|v}degY A

for each v ∈ MF . Of course a similar inequality holds for |B(x, y)|v. These inequalitiesinserted into (A.2.2) conclude this part of the lemma.

Our proof for Part (iii) follows the lines of Proposition 5 in [BM06]: say A =aqY

q + · · · + a0 with ai ∈ Q[X] and aq 6= 0. Because of hypothesis there exists q′ ≥ 1maximal such that aq′(x) 6= 0. Let F be a number field that contains x, y, and thecoefficients of A. If | · | = | · |v is an absolute value on F , then

|aq′(x)yq′ | ≤ δv(q′) max0≤k≤q′−1

{|ak(x)|}max{1, |y|}q′−1

and somax{1, |y|} ≤ δv(q′) max

0≤k≤q′{|ak(x)|/|aq′(x)|}.

We use the last inequality and the product formula to show

(A.2.3) h(y) ≤ log q + [F : Q]−1∑

v∈MF

dv log max0≤k≤q′

{|ak(x)|v}.

By the triangle inequality we have

|ak(x)|v ≤ δv(1 + degX P ) max{1, |x|v}degX P |A|vfor any absolute value on F . We apply this inequality to (A.2.3) to complete theproof. �

We will often apply property (iii) of the previous lemma to a non-zero A ∈ Q[Y ]and x = 0. Inequality (A.2.1) then reduces to h(y) ≤ log degY A + hp(A).

We need to introduce a crude notion of a sparsity: if A = (aij) is an M ×N matrix,then we set

S(A) = max1≤i≤M

|{j; aij 6= 0}|.

If A has algebraic coefficients and is non-zero we define the height hp(A) as the heightof [aij ; 1 ≤ i ≤ M, 1 ≤ j ≤ N ] ∈ PMN−1(Q).

Next we adapt Roy and Thunder’s Absolute Siegel Lemma from [RT96] to ournotation. But first we remark that a version due to David and Philippon (Lemma 4.7in [DP99]) which uses Zhang’s Theorem 5.2 from [Zha95a] would also suffice.


Lemma A.2. Let A = (aij) ∈ MatMN (Q), rankA = M < N . Then there existsv ∈ PN−1(Q) such that Av = 0 and

h(v) ≤ log 22

N

N −M+

M

N −M(log S(A) + hp(A)) +

N −M

4.

Proof. We apply Theorem 1 of [RT99] (cf. [RT96]) which states that there existsv ∈ PN−1(Q) with Av = 0 and

(A.2.4) h(v) ≤ 1N −M

log H(V ) +N −M

4

where V ⊂ QN is the kernel of A and H(V ) is the height of the vector space V as definedin [RT96]. We note that the (logarithm of the) height used in [RT99] and [RT96] usesthe Euclidean norm at infinite places and is hence at least as big as our notion of heightwhich uses the infinity norm at all places. By Theorem 1.1 in [RT96] H(V ) is equal tothe height of the vector space W ⊂ QN spanned by the rows of A. Because the rankof A is M , the rows of A are a basis of W . The height H(W ) is then just the heightof the projective vector composed of the determinants of all M ×M minors of A withthe Euclidean norm taken at the infinite places and maximum norm at the finite places.A simple application of the triangle inequality and the fact that the expansion of eachdeterminant in question involves at most S(A)M non-zero terms gives

log H(V ) = log H(W ) ≤ 12

log(

N

M

)+ M(log S(A) + hp(A)).

The proof follows from this inequality, (A.2.4), and(

NM

)≤ 2N . �

Lemma A.3. Let P ∈ Q[X, Y ] with p = degX P > 0 and q = degY P > 0. Further-more, let m,n be integers with m ≥ q,n ≥ p and define t = q(n+1)−mp. If t ≥ 1, thenthere exist A,B ∈ Q[X, Y ] with

AY m −B ∈ P ·Q[X, Y ]\{0}, degX A, degX B ≤ n, degY A,degY B ≤ q − 1(A.2.5)

and

hp(A,B) ≤ mn

t(log(

√8p) + hp(P )) +

nq

2.

Proof. Let Q =∑

k,l qklXkY l ∈ Z[X, Y, qkl] with degX Q = n − p and degY Q =

m− 1. Define the linear forms fij ∈ Q[qkj ] (0 ≤ i ≤ n, 0 ≤ j ≤ m + q − 1) by

PQ =∑i,j

fijXiY j .

We note that fij is a linear form in the qkj such that its non-zero coefficients arecoefficients of P . So

(A.2.6) fij = 0, for 0 ≤ i ≤ n, q ≤ j ≤ m− 1


is a system of linear equations with rank M in the

N = m(n− p + 1)

variables qij . Clearly one has

(A.2.7) M ≤ (n + 1)(m− q) = N − t.

Because N −M ≥ t ≥ 1 there is a non-trivial solution. Any such solution gives rise toa non-zero polynomial Q ∈ Q[X, Y ] such that the coefficients of PQ satisfy (A.2.6) andhence PQ = AY m−B for unique polynomials A,B ∈ Q[X, Y ] with degX A,degX B ≤ nand degY A,degY B ≤ q − 1.

A peculiarity of Roy and Thunder’s version of Siegel’s Lemma is that the secondterm in upper bound (A.2.4) works against us if the rank M is small. We work out alower bound for M : a non-trivial linear combination of XiY jP where 0 ≤ i ≤ n − pand q ≤ j ≤ m− q − 1 is not of the form AY m −B with A and B satisfying the degreebounds in (A.2.5). Therefore we get a lower bound M ≥ (n− p + 1) max{0,m− 2q}, so

(A.2.8) N −M ≤ (n− p + 1)(m−max{0,m− 2q}) ≤ 2nq.

We will apply Siegel’s lemma to find a solution Q with small height. If M = 0,nothing has to be done as Q = 1 is possible. So say M ≥ 1. We choose a subset of thelinear forms fij (0 ≤ i ≤ n, q ≤ j < m) with rank M . We use the coefficients of eachsuch linear form to define a row in the M ×N matrix A. It was noted that the non-zeroentries of A are coefficients of P , hence hp(A) ≤ hp(P ). Furthermore, it is clear fromthe definition that each fij involves at most 1+min{p, q} non-zero coefficients and henceS(A) ≤ 2p. By Lemma A.2 and our discussion above there exists a non-zero solutionQ ∈ Q[X, Y ] of (A.2.6) that satisfies

hp(Q) ≤ log 22

N

N −M+

M

N −M(log(2p) + hp(P )) +

N −M

4.

Lemma A.1(i) implies hp(PQ) ≤ log(2p) + hp(P ) + hp(Q). Furthermore, we use theinequalities (A.2.7) and (A.2.8) to conclude

hp(PQ) ≤ log(2p) + hp(P ) +log 2

2N

t+

M

t(log(2p) + hp(P )) +

nq

2

≤ M + t

tlog(2p) +

log 22

N

t+

M + t

thp(P ) +

nq

2

≤ N

t(log(

√8p) + hp(P )) +

nq

2.

This inequality completes the proof because N ≤ mn. �

Let A, B, and P be as in the lemma above, then P cannot divide A. Indeedassuming the contrary, then P also divides B. Because degY A, degY B < degY P wehave A = B = 0, a contradiction to AY m − B 6= 0. For similar reasons and sincedegX P > 0, P cannot divide B.


3. Zero bounds

We need only the most basic facts about function fields which we recall here for thereader’s convenience.

Let F be a field which is an extension of an algebraically closed field L. Assumethat there exists an element t ∈ F transcendental over L such that F is a finite fieldextension of L(t). Then F is a function field over L. We define MF to be the set ofthe maximal ideals of all the proper valuation rings of F containing L. This set is thefunction field analogue of MK for a number field K. We will identify an element ofMF with the valuation function it induces. Hence the elements of MF are surjectivemaps v : F ∗ → Z with v(ab) = v(a) + v(b) if a, b ∈ F ∗, v(a + b) ≥ min{v(a), v(b)} ifa, b, a + b ∈ F ∗, and v(λ) = 0 if λ ∈ L∗.

We have the following properties: if a ∈ F ∗ we have v(a) = 0 for all but finitelymany v ∈ MF and ∑

v∈MF

v(a) = 0.

Furthermore, if a ∈ F\L then∑v∈MF

max{0, v(a)} = [F : L(a)].

For the rest of this section P ∈ Q[X, Y ] will, unless stated otherwise, be a fixedirreducible polynomial and F will denote the quotient field of the domain Q[X, Y ]/(P );then F is a function field over L = Q . By abuse of notation we shall consider polyno-mials in Q[X, Y ] as elements of F via the natural map. Note that any polynomial inQ[X, Y ] that is not divisible by P maps to F ∗.

Let π = (x, y) ∈ Q2 with P (π) = 0 such that ∂P∂X , ∂P

∂Y do not both vanish at π, thenwe call π a regular zero of P . Let us assume for the moment π = (x, y) and ∂P

∂Y (π) 6= 0,then there exist a unique vπ ∈ MF with vπ(X −x) = 1 and vπ(Y − y) > 0. There existsE in Q[[T ]], the ring of formal power series, such that E(0) = 0 and P (x+T, y+E) = 0.For any A ∈ Q[X, Y ] not divisible by P we have

ordA(x + T, y + E) = vπ(A)

where ord is the usual valuation on Q[[T ]]. Therefore vπ(A) = 0 if and only if A(x, y) 6=0.

Of course if ∂P∂X (π) 6= 0 then the results in the last paragraph hold with the roles of

X and Y reversed.Proofs for these statements can be found in [Che51].Let A ∈ Q[X, Y ], we define

D(A) =∂P

∂Y

∂A

∂X− ∂P

∂X

∂A

∂Y∈ Q[X, Y ].

We also set D0(A) = A and inductively Ds(A) = D(Ds−1(A)) for a positive integer s.A formal verification gives D(AB) = D(A)B +AD(B) if B is in Q[X, Y ]. Thus we have


Leibniz’s rule:

Ds(AB) =s∑

k=0

( s

k

)Dk(A)Ds−k(B).

Lemma A.4. Let K be a number field and P ∈ K[X, Y ] with p = degX P > 0 andq = degY P > 0. Furthermore, assume v ∈ MK and A ∈ K[X, Y ] with degX A ≤ n,degY A ≤ q. Then for any non-negative s ∈ Z

degX Ds(A) ≤ s(p− 1) + n, degY Ds(A) ≤ s(q − 1) + q,(A.3.1)

and if r = max{p, q} then

|Ds(A)|v ≤ δv(4pq(sr + n))s|P |sv|A|v.(A.3.2)

Proof. We note degX D(A) ≤ p− 1 + degX A and so the first inequality in (A.3.1)follows by induction on s. The second inequality is proved similarly. We now prove(A.3.2) by induction on s. The case s = 0 being trivial we assume s ≥ 1. For brevityset | · | = | · |v. We apply Lemma A.1(i) to show

|Ds(A)| ≤ δv(2(r′ + 1))max{∣∣∣∣∂P

∂Y

∣∣∣∣ ∣∣∣∣∂Ds−1(A)∂X

∣∣∣∣ , ∣∣∣∣ ∂P

∂X

∣∣∣∣ ∣∣∣∣∂Ds−1(A)∂Y

∣∣∣∣}where r′ = min{p, q}. By bounding the partial derivatives of the polynomials in a usualway we get

|Ds(A)| ≤ δv(4r′r)|P ||Ds−1(A)|max{δv(degX Ds−1(A)), δv(degY Ds−1(A))}.

We apply r′r = pq and the inequality (A.3.1) to get

|Ds(A)| ≤ δv(4pq max{(s− 1)(p− 1) + n, (s− 1)(q − 1) + q})|P ||Ds−1(A)|.

The expressions inside “max” are bounded above by sr + n. Applying the inductionhypothesis completes the proof. �

Lemma A.5. Let π = (x, y) ∈ Q2 be a regular zero of P and let v = vπ ∈ MF bethe valuation described above. If A ∈ Q[X, Y ] is not divisible by P and A(π) = 0, thenD(A) is not divisible by P and

v(D(A)) = v(A)− 1.

Proof. We shall assume ∂P∂Y (π) 6= 0, the case ∂P

∂X (π) 6= 0 is similar. There existsE ∈ TQ[[T ]] such that P (x + T, y + E) = 0 and v(A) = ord A(x + T, y + E) ≥ 1. Bythe chain rule we have

(A.3.3) 0 =d

dTP (x + T, y + E) =

∂P

∂X(x + T, y + E) +

dE

dT

∂P

∂Y(x + T, y + E).


We use the definition of D and (A.3.3) to obtain

ordD(A)(x + T, y + E) = ord((

∂P

∂Y

∂A

∂X− ∂P

∂X

∂A

∂Y

)(x + T, y + E)

)= ord

∂P

∂Y(x + T, y + E) + ord

(∂A

∂X(x + T, y + E) +

dE

dT

∂A

∂Y(x + T, y + E)

).

Note that by hypothesis we have

(A.3.4) ord∂P

∂Y(x + T, y + E) = 0.

We insert (A.3.4) into the equality above and use the chain law to get

ordD(A)(x + T, y + E) = ordd

dTA(x + T, y + E) = ordA(x + T, y + E)− 1

hence v(D(A)) = v(A)− 1. In particular P does not divide D(A). �

Lemma A.6. Let A,B, P, m, p, q, t be as in Lemma A.3. Furthermore, assume P isirreducible and deg P = p + q. Let π ∈ Q2 be a regular zero of P , then there exists aninteger s with 0 ≤ s ≤ t + pq − p − q such that Ds(A)(π) 6= 0 and Dk(A)(π) = 0 for0 ≤ k < s.

Proof. For brevity set v = vπ. Clearly X, Y ∈ F\Q since p and q are both positive.Furthermore, A,B 6= 0 in F by the comment after the proof of Lemma A.3. Finallyv(X), v(Y ) ≥ 0.

We first claim that for any v′ ∈ MF at least one of the two v′(X), v′(Y ) is non-negative. Indeed we argue by contradiction so let us assume v′(X) < 0 and v′(Y ) < 0.Then for any integers i, j with 0 ≤ i ≤ p, 0 ≤ j ≤ q and i + j < p + q we have

(A.3.5) iv′(X) + jv′(Y ) > pv′(X) + qv′(Y ).

Now by hypothesis P = αXpY q + P with α 6= 0 and deg P < p + q. We apply theultrametric inequality and (A.3.5) to get

pv′(X) + qv′(Y ) ≥ min0≤i≤p,0≤j≤q

i+j<p+q

{iv′(X) + jv′(Y )} > pv′(X) + qv′(Y ),

a contradiction.Now assume v′ ∈ MF such that v′(Y ) < 0. Then v′(X) ≥ 0 by the discussion above

and

(A.3.6) v′(A) = v′(Y −mB) = −mv′(Y ) + v′(B).

Now degY B ≤ q − 1 as a polynomial so we apply the ultrametric inequality to get

v′(B) ≥ (q − 1)v′(Y ).

We insert this last inequality into (A.3.6) to find

(A.3.7) v′(A) ≥ (q −m− 1)v′(Y ) ≥ −v′(Y ) > 0


because q ≤ m. Hence∑v′∈MF

max{0, v′(A)} ≥ max{0, v(A)}+∑

v′∈MF ,v′(Y )<0

max{0, v′(A)}

≥ v(A) + (m + 1− q)∑

v′∈MF ,v′(Y )<0

max{0,−v′(Y )}(A.3.8)

where the last inequality follows from (A.3.7). Next we insert the equality∑v′∈MF ,v′(Y )<0

max{0, v′(Y −1)} = [F : Q(Y −1)] = [F : Q(Y )] = p

into (A.3.8) to see

(A.3.9)∑

v′∈MF

max{0, v′(A)} ≥ v(A) + (m + 1− q)p > 0.

In particular A /∈ Q.We continue by bounding the left-hand side of (A.3.9) from above. If v′ ∈ MF with

v′(X) ≥ 0 and v′(Y ) ≥ 0 then v′(A) ≥ 0 because A is a polynomial in X and Y . Hence∑v′∈MF

max{0,−v′(A)} ≤∑

v′∈MF ,v′(X)<0

max{0,−v′(A)}+(A.3.10)

∑v′∈MF ,v′(Y )<0

max{0,−v′(A)}.

Actually equality holds above because at most one v′(X), v′(Y ) can be negative, but thisis not important here. Around (A.3.7) we showed that if v′(Y ) < 0 then v′(A) > 0, hencethe second term on the right-hand side of (A.3.10) is zero. Now recall that degX A ≤ n;if v′(X) < 0 then v′(Y ) ≥ 0 and the ultrametric inequality leads us to v′(A) ≥ nv′(X).If we insert this inequality into (A.3.10) we get∑

v′∈MF

max{0,−v′(A)} ≤ n∑

v′∈MFv′(X)<0

max{0,−v′(X)}(A.3.11)

= n[F : Q(X−1)] = n[F : Q(X)] = nq.

The left-hand sides of (A.3.9) and (A.3.11) are both equal to [F : Q(A)], so we get

v(A) ≤ nq −mp + pq − p = t + pq − p− q.

If we set s = v(A), then Lemma A.5 and induction give v(Dk(A)) = v(A) − k for0 ≤ k ≤ s. Hence Ds(A)(x, y) 6= 0 and Dk(A)(x, y) = 0 for 0 ≤ k < s. �


4. Completion of proof

Lemma A.7. Let P ∈ Q[X, Y ] be irreducible with p = degX P > 0 and q = degY P .Then there is a root of unity ξ such that the polynomial P = XpP (X−1 + ξ, Y ) has totaldegree p + q, is irreducible in Q[X, Y ], and satisfies

degX P = p, degY P = q, hp(P ) ≤ hp(P ) + p log 2.

Proof. We may write P =∑

j ajYj with aj =

∑i aijX

i ∈ Q[X]. By hypothesiswe have aq 6= 0 and aq has degree at most p as a polynomial in X. We may choose aroot of unity ξ such that aq(ξ) 6= 0. A direct computation shows that P is irreducible.And by construction degX P ≤ p, degY P ≤ q, therefore deg P ≤ p + q.

Say 0 ≤ i ≤ p and 0 ≤ j ≤ q, then the coefficient of XiY j in P equals

(A.4.1)p∑

k=p−i

akj

(k

i− p + k

)ξi−p+k.

So if i = p and j = q we see that the coefficient of XpY q is non-zero and conclude thatdegX P = p, degY P = q, and deg P = p + q.

Now say K is a number field that contains ξ and the coefficients of P . If v ∈ MK ,then by (A.4.1) and standard facts on binomial coefficients we get

|P |v ≤ δv

p∑k=p−i

(k

i− p + k

) |P |v = δv

((p + 1

i

))|P |v ≤ δv(2p)|P |v.

These local bounds complete the proof. �

Lemma A.8. Let P ∈ Q[X, Y ] be irreducible with p = degX P > 0 and q = degY P >

0. If (x, y) ∈ Q2 with P (x, y) = 0 is not a regular zero of P then

h(y) ≤ 2php(P ) + 5p log(2pq).

Proof. Let D ∈ Q[Y ] be the resultant of the two polynomials P, ∂P∂X ∈ Q(Y )[X]

(cf. [Lan02] page 200). Then D 6= 0 because P is irreducible in Q(Y )[X]. The resultantD is the determinant of a (2p−1)× (2p−1) matrix whose entries, denoted here by mij ,are polynomials in Y with degrees bounded by q. Hence deg D ≤ (2p − 1)q ≤ 2pq. IfK is a number field containing the coefficients of the mij and v ∈ MK , then by LemmaA.1(i)

(A.4.2) |D|v ≤ δv((2p− 1)!)maxσ{|m1,σ(1) · · ·m2p−1,σ(2p−1)|v}

where σ runs over all permutations of the first 2p−1 positive integers. We apply LemmaA.1(i) to bound

|m1,σ(1) · · ·m2p−1,σ(2p−1)|v ≤ δv(2q)2p−2|m1,σ|v · · · |m2p−1,σ(2p−1)|v.This inequality and |mij |v ≤ δv(p)|P |v inserted into (A.4.2) gives

|D|v ≤ δv((2p− 1)!(2pq)2p)|P |2p−1v ≤ δv(4p2q)2p|P |2p−1

v ≤ δv(2pq)4p|P |2p−1v .


Hence hp(D) ≤ 2php(P ) + 4p log(2pq).Now if (x, y) ∈ Q2 with P (x, y) = 0 is not a regular zero of P , then D(y) = 0. By

Lemma A.1(iii) we can bound h(y) ≤ hp(D) + log deg D ≤ hp(D) + log(2pq). The prooffollows from this inequality together with the bound for hp(D). �

Lemma A.9. Let P ∈ Q[X, Y ] be irreducible with p = degX P > 0, q = degY P > 0,and deg P = p + q. If (x, y) ∈ Q2 is a regular zero of P and h(x) ≥ hp(P ), then

h(y) ≤ p

qh(x) + 16p max{log(

√8p)2, hp(P )}1/2 max{1, h(x)}1/2 + 17p log(3q).

Proof. For brevity we set

h = max{1, hp(P )}1/2, k = max{1, h(x)}1/2.

We note k ≥ h and define

(A.4.3) m = pq2

[k

h

], n = m

p

q+ p− 1.

Then m and n are integers with m ≥ q, n ≥ p. Let t be as in Lemma A.3, we havet = pq ≥ 1. Now let A,B ∈ Q[X, Y ] be as in Lemma A.3. Because of Lemma A.6 thereexists an integer s with 0 ≤ s ≤ t + pq− p− q ≤ 2pq− 1 such that Ds(A)(x, y) 6= 0 andDk(A)(x, y) = 0 for all 0 ≤ k < s. We apply Leibniz’s rule to Ds(AY m − B) and theuse the fact that AY m −B is divisible by P to conclude

ym =Ds(B)(x, y)Ds(A)(x, y)

.

An application of Lemma A.1(ii) gives

mh(y) ≤ hp(Ds(A), Ds(B)) + max{degX Ds(A),degX Ds(B)}h(x)(A.4.4)

+ max{degY Ds(A),degY Ds(B)}h(y)

+ log max{(1 + degX Ds(A))(1 + degY Ds(A)),

(1 + degX Ds(B))(1 + degY Ds(B))}.

Lemma A.4 applied to A and B implies

degX Ds(A),degX Ds(B) ≤ sp + n,

degY Ds(A),degY Ds(B) ≤ q(s + 1), and

hp(Ds(A), Ds(B)) ≤ hp(A,B) + shp(P ) + s log(4pq(n + sr)).

The last line follows from summing up the local bounds in (A.3.2). We insert thesebounds in (A.4.4) to see

mh(y) ≤ hp(A,B) + shp(P ) + s log(4pq(n + sr)) + (sp + n)h(x) + q(s + 1)h(y)

+ log(sp + n + 1)(sq + q + 1).


Next we use the bound given for hp(A,B) in Lemma A.3 to get

mh(y) ≤ mn

t(log(

√8p) + hp(P )) +

nq

2+ shp(P ) + s log(4pq(n + sr))(A.4.5)

+ (sp + n)h(x) + q(s + 1)h(y) + log(sp + n + 1)(sq + q + 1).

To control h(y) on the right side of (A.4.5) we apply Lemma A.1(iii) to P ; we obtain

h(y) ≤ ph(x) + hp(P ) + log(2pq).

We insert the inequality above into (A.4.5) and see

mh(y) ≤mn

t(log(

√8p) + hp(P )) +

nq

2+ shp(P ) + s log(4pq(n + sr))

+ (sp + n + pq(s + 1))h(x) + q(s + 1)hp(P ) + q(s + 1) log(2pq)

+ log(sp + n + 1)(sq + q + 1).

We recall h(x) ≤ k2 and hp(P ) ≤ h2. By collecting the hp(P )’s, h(x)’s and applyingnm ≤ p

q + pm , which holds by (A.4.3), we get

h(y) ≤(

n

t+

s + q(s + 1)m

)h2 +

p(s + 1) + pq(s + 1)m

k2 +p

qh(x) +

n

tlog(

√8p)

(A.4.6)

+nq

2m+

s

mlog(4pq(n + sr)) +

q(s + 1)m

log(2pq)

+1m

log(sp + n + 1)(sq + q + 1).

We now continue by bounding each term in the right-hand side of inequality (A.4.6).The elementary inequalities 1

2pq2 kh ≤ m ≤ pq2 k

h and m ≥ pq2, which hold because ofour assumption k ≥ h, will be used throughout the process.

Our definition in (A.4.3) and t = pq imply

(A.4.7)n

t≤ m

q2+

1q.

Furthermore, we use s ≤ 2pq − 1 to obtain a bound for the first term on the right of(A.4.6) (

n

t+

s + q(s + 1)m

)h2 ≤

(m

q2+

1q

+4pq2

m

)h2 ≤ (p

k

h+

1q

+ 4)h2 ≤ 6phk.(A.4.8)

The second term in (A.4.6) can bounded as follows

(A.4.9)p(s + 1) + pq(s + 1)

mk2 ≤ 4p2q2

mk2 ≤ 8phk.

We skip the third term, which is the main contribution to the height of y, and boundthe fourth with the help of (A.4.7):

(A.4.10)n

tlog(

√8p) ≤ (

m

q2+

1q) log(

√8p) ≤ (p

k

h+

1q) log(

√8p) ≤ 2p

k

hlog(

√8p).


We use nm ≤ p

q + pm again and also m ≥ q to bound the fifth term:

(A.4.11)nq

2m≤ q

2(p

q+

p

m) =

p

2+

pq

2m≤ p.

For the sixth term we use n + sr ≤ n + 2pqr ≤ 3p2q2n to get

s

mlog(4pq(n + sr)) ≤ 2

pq

mlog(12p3q3n) = 2

pq

mlog(12p3q3) + 2pq

log n

m(A.4.12)

≤ 8log(2pq)

q+ 2pq

log n

m.

For positive α and β we have log(α+β) ≤ αβ +log β. So log n ≤ log(mp

q +p) ≤ mq +log p

with α = mpq and β = p. Therefore pq log n

m ≤ p + log p. We apply this inequality andlog q

q ≤ 1e , here e = 2.71828 . . . , to (A.4.12) and get

(A.4.13)s

mlog(4pq(n + sr)) ≤ 8 log(2p) +

8e

+ 2p + 2 log p.

The second to last term in (A.4.6) can be bounded as

(A.4.14)q(s + 1)

mlog(2pq) ≤ 2pq2

mlog(2pq) ≤ 2 log(2pq).

Finally, to tackle the last term we use n ≤ 2pm and p ≤ m which follow from(A.4.3), so log n

m ≤ log(2pm)m ≤ log(2p)

p + log mm ≤ 2

e + 1e by elementary calculus. Hence with

s ≤ 2pq

1m

log(sp + n + 1)(sq + q + 1) ≤ 1m

log(2p2q + n + 1)(2pq2 + q + 1)(A.4.15)

≤ 1m

log(16p3q3n) ≤ log 16 + 3log p

p+ 3

log q

q2+

log n

m

≤ log 16 +3e

+32e

+3e

<285

.

In the second to last inequality we used log qq2 ≤ 1

2e .The sum of the right-hand sides in (A.4.11), (A.4.13), (A.4.14), and (A.4.15) can be

bounded with elementary calculus and the fact that p ∈ N:

p + 8 log(2p) +8e

+ 2p + 2 log p + 2 log(2pq) +285≤ 13p + 2 log q +

285

.

It is easy to show 13p + 2 log q + 28/5 ≤ 17p log(3q) by considering the two cases p = 1and p ≥ 2 separately.

We use this estimate together with (A.4.8), (A.4.9), and (A.4.10) to bound (A.4.6)from above as follows:

h(y) ≤ p

qh(x) + 14phk + 2p

k

hlog(

√8p) + 17p log(3q).


If we abbreviate h′ = max{(log(√

8p))2, hp(P )}1/2 ≥ h, then kh log(

√8p) ≤ h′k and so

h(y) ≤ p

qh(x) + 16ph′k + 17p log(3q)

as required. �

Proposition A.1. Let P ∈ Q[X, Y ] be irreducible with p = degX P > 0 and q =degY P > 0. If (x, y) ∈ Q2 with P (x, y) = 0, then

h(y) ≤ p

qh(x) + 28p max{p, hp(P )}1/2 max{1, hp(P ), h(x)}1/2 + 18p log(3q).

Proof. If h(x) < hp(P ) + 2p log 2, then Lemma A.1(iii) applied to P gives

h(y) ≤ ph(x) + log(2pq) + hp(P )

≤ p(hp(P ) + 2p log 2)1/2h(x)1/2 + log(2pq) + hp(P ).

This inequality is clearly stronger than the assertion.So we will assume

(A.4.16) h(x) ≥ hp(P ) + 2p log 2.

Now if both partial derivatives of P vanish at (x, y), then Lemma A.8 and log p ≤ p1/2

giveh(y) ≤ 2php(P ) + 5p log(2pq) ≤ 2php(P ) + 5p3/2 + 5p log(2q),

which is also stronger than our assertion. From now on we assume that (x, y) is a regularzero of P .

There exist ξ ∈ Q and P as in Lemma A.7. If x = ξ or x = 0, then h(x) = 0,but this contradicts (A.4.16). Hence x 6= 0, ξ and ((x− ξ)−1, y) is a zero of P . A shortcalculation shows that this zero is regular. The height inequalities

h((x− ξ)−1) = h(x− ξ) ≥ h(x)− h(ξ)− log 2 = h(x)− log 2,

hp(P ) ≤ hp(P ) + p log 2,(A.4.17)

and (A.4.16) imply h((x− ξ)−1) ≥ hp(P ). We may thus apply Lemma A.9 to the point((x− ξ)−1, y). We use h((x− ξ)−1) ≤ h(x) + log 2 and again (A.4.17) to show

h(y) ≤p

qh(x) +

p

qlog 2(A.4.18)

+ 16p max{(log(√

8p))2, hp(P ) + p log 2}1/2 max{1, h(x) + log 2}1/2

+ 17p log(3q).

We note (log(√

8p))2 ≤ 1.6p ≤ (1 + log 2)p since p ≥ 1 and so

max{(log(√

8p))2, hp(P ) + p log 2} ≤ (1 + log 2) max{p, hp(P )}.Furthermore, we have max{1, h(x) + log 2} ≤ (1 + log 2)max{1, h(x)}. The propositionfollows from these last two inequalities, (A.4.18), and p

q log 2 ≤ p log(3q).�


Proof of Theorem A.1: For brevity say r = max{p, q}. If max{h(x), h(y)} < hp(P ),then clearly

|ph(x)− qh(y)| ≤ (p + q) max{h(x), h(y)} ≤ 2pqhp(P )1/2 max{h(x), h(y)}1/2

and the theorem follows in this case.So let us assume max{h(x), h(y)} ≥ hp(P ). We note that by symmetry Proposition

A.1 implies

|ph(x)− qh(y)| ≤ 28pq max{r, hp(P )}1/2 max{1, h(x), h(y)}1/2 + 18pq log(3r).

The Theorem clearly follows from this inequality and 18 log(3r) ≤ 23r1/2. �

Corollary A.1 is easy to prove with the Quasi-equivalence Theorem.Say P (x, y) = 0 with x, y ∈ Q, h(y) > 2p

q h(x), and h(x) ≥ 1. If h(x) ≤ h(y), thenby Theorem A.1

12h(y) < h(y)− p

qh(x) ≤ 51p max{p, q, hp(P )}1/2h(y)1/2.

Hence the height of y is less then the right of (A.1.3) and therefore so is the height ofx. On the other hand if h(y) ≤ h(x) we get

p

qh(x) < h(y)− p

qh(x) ≤ 51p max{p, q, hp(P )}1/2h(x)1/2.

The resulting bound for h(x) is less than the right side of (A.1.3). �

APPENDIX B

Applications of the Quasi-equivalence Theorem

In this second appendix we show how to use the Quasi-equivalence Theorem from theprevious appendix to deduce explicit versions of four number theoretic results: Theoremsof Bombieri, Masser, and Zannier, of Runge, of Skolem, and of Sprindzhuk. The goalis not to get optimal dependency in the various quantities, but rather to show howTheorem A.1 is connected to these four theorems. All four theorems mentioned areproved using the same auxiliary function constructed in section 2 of appendix A.

1. On a Theorem of Bombieri, Masser, and Zannier

In this first section we give a completely explicit proof of Theorem 1 of [BMZ99]in the two dimensional case. The proof uses Theorem A.1 and closely mimics Bombieri,Masser, and Zannier’s “alternative proof” given in §3 of [BMZ99]. In fact the oc-currence of a quasi-equivalence of heights statement in [BMZ99] was the main drivebehind proving Theorem A.1.

Bombieri, Masser, and Zannier’s Theorem has already been discussed in chapter 6.

Theorem B.1. Let P ∈ Q[X, Y ] be irreducible with p = degX P , q = degY P , andnot of the form αXpY q−β or αXp−βY q. If P (x, y) = 0 with x, y ∈ Q∗ multiplicativelydependent, then

max{h(x), h(y)} ≤ 3.105(deg P )3 max{pq, hp(P )}.

We recall that x and y are multiplicatively dependent if and only if (x, y) is containedin a proper algebraic subgroup of Gn

m.We need a preparatory lemma.

Lemma B.1. Let P ∈ Q[X, Y ] be irreducible with p = degX P > 0 and q = degY P >0. If b ∈ N, there exists Q ∈ Q[X, Y ] irreducible with

(B.1.1) 1 ≤ degX Q ≤ bp, 1 ≤ degY Q ≤ q, and hp(Q) ≤ bhp(P ) + 2b(p + q) log 2

such that P divides Q(X, Y b).

Before we prove Lemma B.1 we recall two height inequalities for polynomials: letf1, . . . , fn ∈ Q[X, Y ] be non-zero with f = f1 · · · fn and d = degX f + degY f , then

(B.1.2) −d log 2 +n∑

i=1

hp(fi) ≤ hp(f) ≤ d log 2 +n∑

i=1

hp(fi).

119


A proof not restricted to the two variable case is given in [BG06] (Theorem 1.6.13 page28).Proof of Lemma B.1: Let ζ be a primitive bth root of unity and set A =

∏bi=1 P (X, ζiY ).

Since A(X, Y ) = A(X, ζY ) we have A(X, Y ) = B(X, Y b) for some B ∈ Q[X, Y ].Clearly degX B = bp, degY B = q, and hp(B) = hp(A). With (B.1.2) and hp(P ) =hp(P (X, ζiY )) we may bound

(B.1.3) hp(B) ≤ bhp(P ) + b(p + q) log 2.

Also P divides B(X, Y b) by construction. But our B need not be irreducible, never-theless B has an irreducible factor Q such that P divides Q(X, Y b). The degree upperbounds in (B.1.1) hold and so do the lower bounds since pq > 0. Furthermore, theheight bound in (B.1.1) follows from (B.1.2) and (B.1.3). �

Proof of Theorem B.1: Let P be as in the hypothesis, we have p, q > 0. By symmetrywe may assume h(x) ≥ h(y) and without loss of generality also h(x) ≥ 1. There existintegers r, s not both zero with xrys = 1. We will often use the fact that |r|h(x) = |s|h(y)which follows from chapter 1. If s = 0, then r 6= 0 and so h(x) = 0, contradicting ourassumption. Hence s 6= 0 and we may find t ∈ Q∗ with x = ts, y = ζt−r for some rootof unity ζ. We define θ = r/s, then |θ| = h(y)/h(x) ≤ 1.

Let B > 1, by [Cas57] page 1, there exist integers a, b with 1 ≤ b < B and|θb− a| ≤ B−1. We take B = 2q and define γ = xayb = ζbtas−br. Therefore

(B.1.4) h(γ) = |as− br|h(t) = |θb− a|h(x) ≤ h(x)B

< h(x).

Let Q be the polynomial from Lemma B.1. We set R = Q(X, X−aY )Xw wherew is an integer such that R is a polynomial not divisible by X. We have R(x, γ) =Q(x, yb) = 0 since P divides Q(X, Y b).

We continue by comparing h(x) and h(γ) with Theorem A.1. It is easy to see thatR is irreducible. We define m = degX R and n = degY R. Bounding n is not difficult:as n = degY Q we obtain

1 ≤ n ≤ q.

Since |θb − a| ≤ B−1 we have |a| ≤ B−1 + |θ|b ≤ 1 + b < 1 + B = 1 + 2q, so |a| ≤ 2q.With the bound for degX Q and degY Q in Lemma B.1 we have

m ≤ 2 max{1, |a|}deg Q ≤ 4q(bp + q) ≤ 4q2(2p + 1) ≤ 12pq2.

Finally we show m ≥ 1. Indeed assume m = 0, then R has degree 1 in Y by irreducibility.Therefore P divides a polynomial of the form αXuY v−β or αXu−βY v. But then P isalso of this form, contradicting the hypothesis. Since R and Q have the same coefficientswe get

(B.1.5) hp(R) = hp(Q) ≤ bhp(P ) + 2b(p + q) log 2 ≤ 2qhp(P ) + 4q(p + q) log 2

by Lemma B.1.

B. APPLICATIONS OF THE QUASI-EQUIVALENCE THEOREM 121

Having bounded height and partial degrees of R and since R(x, γ) = 0 we applyTheorem A.1 to deduce

h(x) ≤ n

mh(γ) + 51n max{m,n, hp(R)}1/2h(x)1/2

≤ qh(x)B

+ 51q max{12pq2, 2qhp(P ) + 4q(p + q) log 2}1/2h(x)1/2,

in the second inequality we used (B.1.4) and (B.1.5). Since qB = 1

2 we conclude12h(x)1/2 ≤ 51

√12q3/2 max{pq, hp(P ) + pq}1/2 ≤ 250q3/2 max{pq, hp(P )}1/2.

This inequality completes the proof. �

2. On a Theorem of Runge

In [Run87] Runge proved that P (x, y) = 0 admits only finitely many solutionsx, y ∈ Z if P ∈ Q[X, Y ] is irreducible and satisfies the following condition: we havedegX P = degY P = deg P and the homogeneous part of P with maximal degree is nota scalar times the power of an irreducible polynomial in Q[X, Y ]. In fact Runge proveda finiteness result under a weaker hypothesis on P . Runge’s method is effective andseveral explicit bounds for max{|x|, |y|} have been determined: for example Hilliker andStraus in [HS83] or Walsh in [Wal92]. Many approaches use Eisenstein’s Theorem onbounding the coefficients in power series of algebraic functions.

In this section we will prove a simple, explicit version of Runge’s Theorem usingthe Quasi-equivalence Theorem proved in appendix A. We state the main result of thesection:

Theorem B.2. Let P ∈ Z[X, Y ] be irreducible in Q[X, Y ] and assume d = degX P =degY P = deg P . Furthermore, assume that the homogeneous part of P with degree dis not a scalar times the power of an irreducible polynomial in Q[X, Y ]. If P (x, y) = 0with x, y ∈ Z, then

(B.2.1) log max{1, |x|, |y|} ≤ 105d5 max{d, hp(P )}.

A more sophisticated version of Runge’s Theorem, similar to the one of Bombieri([Bom83] page 304), can also be proved using our Quasi-equivalence Theorem.

For the rest of this section let P =∑

i,j pijXiY j ∈ Z[X, Y ] be irreducible in Q[X, Y ]

with d = deg P = degX P = degY P . Furthermore, since Theorem B.2 applies only topolynomials of degree at least 2, we assume d ≥ 2. We use Pd =

∑i+j=d pijX

iY j todenote the homogeneous part of degree d. Finally we can factor Pd as pd0

∏ds=1(X−tsY )

with ts ∈ Q∗.For any s, the function X − tsY on the curve defined by P has degree strictly less

than d, the degree of the function X:

Lemma B.2. Say t = ts. We have h(t) ≤ hp(P )+ log d. Furthermore, if Rt(X, Z) =P (X, t−1(X − Z)) ∈ Q[X, Z], then degZ R = d, degX R ≤ d − 1, and hp(Rt) ≤d(2hp(P ) + log(2d)).


Proof. The bound for h(t) follows from Lemma A.1(iii); indeed t is a zero ofPd(X, 1).

Let i, j ∈ Z, the coefficients of XiZj in Rt is

(−1)jd−j∑k=0

(j + k

k

)pi−k,j+kt

−j−k.

The coefficient of Zd is nonzero and that of XiZj is zero provided i ≥ d. The lemmanow follows from elementary inequalities as in the proof of Lemma A.7. �

By the previous lemma and the Quasi-equivalence Theorem, the height h(x− ty) issmall compared to h(x). We can say even more:

Lemma B.3. Let P (x, y) = 0 with x, y ∈ Z and

(B.2.2) h(x) ≥ 2.104d2 max{d, hp(P )},

then h(y) ≤ 2h(x). Furthermore, if t = ts for some 1 ≤ s ≤ d, then

log max{1, |x− σ(t)y|} ≤ d− 1d

log |x|+ 177d3/2 max{d, hp(P )}1/2√

log |x|

for some embedding σ : Q(t) → C.

Proof. The inequality h(y) ≤ 2h(x) follows from Corollary A.1 in appendix A.Let R = Rt be as in Lemma B.2, then R is irreducible. Now degX R ≥ 1, indeed

if degX R = 0, then by construction P has degree 1, which contradicts our hypothesisd ≥ 2. Let z = x− ty, so R(x, z) = 0. Theorem A.1, degX R ≤ d − 1, and degZ R = dimply

(B.2.3) h(x− ty) = h(z) ≤ d− 1d

h(x) + 51d max{d, hp(R)}1/2 max{1, h(x), h(z)}1/2.

We have the elementary bound h(z) ≤ log 2 + h(x) + h(y) + h(t) which yields h(z) ≤log(2d)+hp(P )+h(x)+h(y) ≤ log(2d)+hp(P )+3h(x) by Lemma B.2 and h(y) ≤ 2h(x).Therefore h(z) ≤ 4h(x), in view of the lower bound (B.2.2). By Lemma B.2 we alsoconclude max{d, hp(R)} ≤ 3d max{d, hp(P )}. With (B.2.3) we obtain

h(x− ty) ≤ d− 1d

h(x) + 177d3/2 max{d, hp(P )}1/2h(x)1/2.

By the definition of the height (cf. chapter 1) there exists an embedding σ : Q(t) → Cwith log max{1, |x − σ(t)y|} ≤ h(x − ty). The lemma follows from this statement andh(x) = log |x|. �

We now prove Theorem B.2:Let P be as in the hypothesis. By assumption there are non-conjugated t′, t′′ ∈ Q∗

that are zeros of Pd(X, 1). Say x and y are as in the hypothesis and |x| ≥ |y|. We mayassume that x satisfies (B.2.2).


Let σ′, σ′′ be the embeddings given by Lemma B.3 applied to t′, t′′ respectively, thenσ′(t′) 6= σ′′(t′′). For brevity we define ξ = x− σ′(t′)y and η = x− σ′′(t′′)y; we eliminatey to get

x =ξσ′′(t′′)− ησ′(t′)σ′′(t′′)− σ′(t′)

.

So

(B.2.4) |x| ≤ 2 max{|ξ|, |η|}max{|σ′(t′)|, |σ′′(t′′)|}|σ′(t′)− σ′′(t′′)|−1

for the complex absolute value | · |.We will bound |ξ| and |η| using Lemma B.3, but first we bound the remaining

absolute values in (B.2.4) using height inequalities: if α ∈ Q, then log max{1, |α|v} ≤[Q(α) : Q]h(α) for any v ∈ MQ(α). This inequality follows immediately from thedefinition of the height. Since for example [Q(t′) : Q] ≤ d and [Q(t′, t′′) : Q] ≤ d2 weget a bound for (B.2.4):

(B.2.5) log |x| ≤ log 2 + d max{h(t′), h(t′′)}+ d2h(σ′(t′)− σ′′(t′′)) + log max{1, |ξ|, |η|}.Let h = max{d, hp(P )}. Lemma B.2 implies the upper bounds

max{h(t′), h(t′′)} ≤ hp(P ) + log d.

Next we apply Lemma B.3 to bound |ξ| and |η|. Together with (B.2.5) and standardheight inequalities we have

log |x| ≤ log 2 + d(log d + hp(P )) + d2(log 2 + 2 log d + 2hp(P )) +d− 1

dlog |x|

+ 177d3/2h1/2√

log |x|

≤d− 1d

log |x|+ 8d2h + 177d3/2h1/2√

log |x|.

Hencelog |x| ≤ 8d3h + 177d5/2h1/2

√log |x|.

If B,C, T ≥ 0 with T ≤ B√

T + C, then

(B.2.6) T ≤ (1 +√

2)2 max{(B/2)2, C} ≤ (1 +√

2)2((B/2)2 + C).

We use (B.2.6) with B = 177d5/2h1/2, C = 8d3h, and T = log |x| to conclude thatlog |x| is at most half the right side of (B.2.1). The upper bound for log |y| follows fromLemma B.3 which implies log max{1, |y|} ≤ 2 log |x|.

3. On a Theorem of Skolem

In 1929 Skolem showed that if P ∈ Z[X, Y ] is irreducible in Q[X, Y ] and P (0, 0) = 0,then there are only finitely many coprime x, y ∈ Z with P (x, y) = 0. Of course Siegel’sTheorem on the finiteness of the integral points on a curve of genus at least 1 super-sedes this result in many cases. Contrary to Siegel’s Theorem, which remains ineffective,Skolem’s Theorem is effective. For example in [Wal92] Walsh proved an effective andexplicit version of Skolem’s Theorem. In [Pou04] Poulakis improved on Walsh’s bound


under some additional restrictions to P . In [Abo06] Abouzaid extended Skolem’s Theo-rem in two ways: first, he considered polynomials P with algebraic coefficients. Second,he generalized the definition of the greatest common divisor, or gcd for short, from pairsof integers to pairs of algebraic numbers. Say P ∈ Q[X, Y ] is irreducible with degY P > 0such that (0, 0) is a non-singular point on the curve defined by P . If P (x, y) = 0 withx, y algebraic and not both zero, then Abouzaid proved that log gcd(x, y) is asymptoti-cally equal to 1

degY P h(x). He also explicitly bounded the difference of these two values.In Theorem B.3 we give an independent proof of Abouzaid’s result based on the workin appendix A on quasi-equivalence of heights.

Let x, y ∈ K where K is a number field and assume that x and y are not both zero.We follow Abouzaid and define

lgcd(x, y) =1

[K : Q]

∑v∈MK

[Kv : Qv] log min{max{1, |x|−1v },max{1, |y|−1

v }}.

Here lgcd(x, y) = h(y) if x = 0 and y 6= 0. Just like the height, lgcd(x, y) does notdepend on the field K containing x and y. For a, b ≥ 0 we have the identity

max{min{1, a},min{1, b}} =max{a, b}

max{1, a, b},

and so we may rewrite lgcd(x, y) as

(B.3.1) lgcd(x, y) = h(x, y)− h([x : y]) ≥ 0.

Here (x, y) ∈ Q2 and [x : y] ∈ P1(Q). We recall our notion of height defined in chapter1 corresponding to these two cases. Finally it is readily checked that lgcd(x, y) =log gcd(x, y) if x and y are integers.

For any non-zero polynomial P (X, Y ) in two variables we define

e(P ) = max{f ∈ Z; Xf |P (X, XY )} ∈ [0,deg P ].

If P =∑

i,j pijXiY j , then e(P ) = min{i + j; pij 6= 0}. Clearly e(P ) is positive if and

only if P (0, 0) = 0. Furthermore, e(P ) = 1 precisely when ∂P∂X (0, 0) 6= 0 or ∂P

∂Y (0, 0) 6= 0.That is when (0, 0) is a non-singular point on the curve defined by P .

We will use Theorem A.1 to give an independent proof of Abouzaid’s Theoremwith slightly improved dependency on the degree of P and without the non-singularityhypothesis. Abouzaid proved his theorem without referring to a quasi-equivalence ofheights result, in fact he uses it to prove variant of our Theorem A.1.

Theorem B.3. Let P ∈ Q[X, Y ] be irreducible with p = degX P > 0, q = degY P >0, and d = deg P . If P (x, y) = 0 where x and y are non-zero algebraic numbers, then

max{∣∣∣∣lgcd(x, y)− e(P )

qh(x)

∣∣∣∣ , ∣∣∣∣lgcd(x, y)− e(P )p

h(y)∣∣∣∣}(B.3.2)

≤ 183d max{d, hp(P )}1/2 max{1, h(x), h(y)}1/2.


Deriving a slightly weaker version of Theorem A.1 from inequality (B.3.2) is simple ifP (0, 0) = 0. In this case the difference |h(x)

q − h(y)p | is at most twice the right-hand side of

(B.3.2). The case P (0, 0) 6= 0 can be handled by replacing P (X, Y ) with P (X +ξ, Y +η)where P (ξ, η) = 0

As a corollary we get a bound for the heights of x and y in terms of lgcd(x, y) assoon as P has no constant term.

Corollary B.1. Let P ∈ Q[X, Y ] and d be as in Theorem B.3. We assume e(P ) ≥1, that is P (0, 0) = 0. If P (x, y) = 0 where x and y are non-zero algebraic numbers,then

(B.3.3) max{h(x), h(y)} ≤ 6d

e(P )lgcd(x, y) + 5.104 d4

e(P )2max{d, hp(P )}.

In particular h(x) and h(y) are bounded if lgcd(x, y) = 0.We prove Theorem B.3 in a series of elementary lemmas and then apply Theorem

A.1.Throughout the remainder of this section P ∈ Q[X, Y ] will be irreducible with

p = degX P > 0, q = degY P > 0, and d = deg P .

Lemma B.4. There exists a root of unity λ such that Q(X, Z) = P (X, Z − λX) isirreducible and satisfies deg Q = degX Q = d, degZ Q = q, and hp(Q) ≤ hp(P ) + q log 2.

Proof. The proof is very similar to the proof of Lemma A.7, so we omit it. �

Lemma B.5. Let x, y ∈ Q with P (x, y) = 0, then

|qh(x, y)− dh(x)| ≤ 110dq max{d, hp(P )}1/2 max{1, h(x), h(y)}1/2.

Proof. Let λ and Q be as in Lemma B.4. For brevity we define h = max{d, hp(P )}and k = max{1, h(x), h(y)}. We set z = λx+y and note Q(x, z) = 0. Let K be a numberfield containing the coefficients of Q and the algebraic numbers λ, x, and y. Let v ∈ MK ,the inequality

δv(2)−1 ≤ max{1, |x|v, |y|v}max{1, |x|v, |z|v}

≤ δv(2)

implies

(B.3.4) |h(x, y)− h(x, z)| ≤ log 2.

We continue by comparing h(x, z) with dq h(z). Say Q =

∑i,j qijX

iY j , then qd0 6= 0.We claim

(B.3.5) |x|v ≤ δv(3d2)|Q|v|qd0|v

max{1, |z|v}.

Indeed, let us assume (B.3.5) is false. In particular |x|v > 1. Let i + j ≤ d with i, jnon-negative integers such that (i, j) 6= (d, 0) and qij 6= 0. If j = 0, then d ≥ i + 1 andso |qd0x

d|v ≥ |qd0xi|v|x|v > δv(3d2)|Q|v|xi|v by the negation of (B.3.5). If j > 0, then


|qd0xd|v ≥ |qd0|v|x|iv|x|

jv > δv(3d2)|Q|v|x|iv|z|

jv, again using the negation of (B.3.5). In

any case we have

|qd0xd|v > δv(3d2)|qijx

izj |v if (i, j) 6= (d, 0).

But since Q(x, z) = 0 and because Q has no more than 12(d + 1)(d + 2) ≤ 3d2 non-zero

coefficients we have a contradiction.Therefore B.3.5 holds for all v ∈ MK , and even with |x|v replaced by the possibly

larger max{1, |x|v, |z|v}. Passing over to heights we have

0 ≤ h(x, z)− h(z) ≤ log(3d2) + hp(Q),here the lower bounds is obvious. We combine this inequality with (B.3.4) and thebound for hp(Q) from Lemma B.4 to obtain

(B.3.6) |h(x, y)− h(z)| ≤ log(6d22q) + hp(P ) ≤ log(6d22d) + hp(P ) ≤ 4h.

The last inequality used log(6d22d) ≤ 3d.We have max{d, hp(Q)} ≤ (1 + log 2)h since hp(Q) ≤ hp(P ) + q log 2. Furthermore,

since h(z) ≤ h(x)+h(y)+log 2 by (1.1.6) in chapter 1 we conclude max{1, h(x), h(z)} ≤(2 + log 2)k. By Theorem A.1 applied to Q we get |dh(x) − qh(z)| ≤ 109dqh1/2k1/2.This inequality and (B.3.6) together imply

|qh(x, y)− dh(x)| ≤ 109dq(hk)1/2 + 4qh.

The lemma now follows easily if k ≥ 16h. If k < 16h, then

|qh(x, y)− dh(x)| ≤ 3dk ≤ 12d(hk)1/2

completes the proof. �

Lemma B.6. Let P (x, y) = 0 with x, y ∈ Q not both zero, then

|(d− e(P ))h(x)− qh([x : y])| ≤ 73dq max{d, hp(P )}1/2 max{1, h(x), h(y)}1/2.

Proof. We may assume x 6= 0 and in this case we have h([x : y]) = h(y/x) bythe product formula. Let e = e(P ) ≤ d. If e = d, then by irreducibility P must behomogeneous of degree 1. The lemma holds in this case as then hp(P ) = h([x : y]) ≤h(x) + h(y). So let us assume e < d.

We define F (U, V ) = U−eP (U,UV ) ∈ Q[U, V ], then F is irreducible with degU F =d− e, degV F = q, and hp(F ) = hp(P ). We have F (x, y/x) = 0 and so by Theorem A.1:

|(d− e)h(x)− qh(y/x)| ≤ 51dq max{d, hp(P )}1/2 max{1, h(x), h(y/x)}1/2.

The lemma now follows from h(y/x) ≤ h(x) + h(y). �

Proof of Theorem B.3: Let P (x, y) = 0 with x, y ∈ Q not both zero. As an immediateconsequence of Lemmas B.5, B.6, and (B.3.1) we get

|lgcd(x, y)− e(P )q

h(x)| ≤ 183d max{d, hp(P )}1/2 max{1, h(x), h(y)}1/2.


This inequality bounds the first quantity on the left in (B.3.2). The bound for the secondquantity follows by considering Q(X, Y ) = P (Y, X); indeed we have e(P ) = e(Q). �

Proof of Corollary B.1: By symmetry we may assume h(x) ≥ h(y) and by (B.3.3) alsoh(x) ≥ 1. Theorem B.3 and q ≤ d imply

h(x) ≤ d

e(P )lgcd(x, y) + 183

d2

e(P )max{d, hp(P )}1/2h(x)1/2.

The upper bound for h(x) in (B.3.3) follows from (B.2.6) with T = h(x), B =183 d2

e(P ) max{d, hp(P )}1/2, and C = de(P ) lgcd(x, y). The same upper bound holds for

h(y) since h(y) ≤ h(x). �

4. On a Theorem of Sprindzhuk

Let P ∈ Q[X, Y ] be an irreducible polynomial with degY P > 0. A famous theoremof Hilbert states that P (x, Y ) ∈ Q[Y ] is irreducible for infinitely many integers x.

In [Spr79b] and [Spr79a] Sprindzhuk proved that if P (0, 0) = 0 and ∂P∂Y (0, 0) 6= 0

then P (l, Y ) ∈ Q[Y ] is irreducible for a sufficiently large prime l. His Theorem iseffective and even holds if l is allowed to be a prime power. In [BM06] Bilu and Massergave a quick proof of a more advanced version of this result, originally also due toSprindzhuk.

In this section we prove a quantitative version of Sprindzhuk’s result using ourQuasi-equivalence Theorem and its consequence Theorem B.3.

Theorem B.4. Let P ∈ Q[X, Y ] be irreducible with degY P > 0 and e(P ) = 1. Ifd = deg P and l ∈ N is a prime with

(B.4.1) log l ≥ 7.104d6 max{d, hp(P )},then P (l, Y ) ∈ Q[Y ] is irreducible.

The condition e(P ) = 1 is equivalent to P (0, 0) = 0 and ( ∂P∂X , ∂P

∂Y )(0, 0) 6= 0. Ourhypothesis is thus slightly weaker than Sprindzhuk’s: he assumes the non-vanishing of∂P∂Y at (0, 0).

In the proof of Theorem B.4 we need the following simple remark: let l ∈ N be aprime and y ∈ Q such that lgcd(l, y) > 0, then

(B.4.2) [K : Q]lgcd(l, y) ≥ log l,

with K = Q(y). Indeed, by definition we have

(B.4.3) [K : Q]lgcd(l, y) =∑

v∈MK

[Kv : Qv] logmax{1, |l|v, |y|v}max{|l|v, |y|v}

.

One of the terms on the right side of the expression above must be positive. Therefore|l|v < 1 and |y|v < 1 for some v ∈ MK . In particular this v is a finite place. Since[Kv : Qv] is the product of ramification index and rest-class residue of the prime idealcorresponding to v we have −[Kv : Qv] log max{|l|v, |y|v} ≥ log l. So (B.4.2) followssince all terms of the sum in (B.4.3) are non-negative.

Proof of Theorem B.4: The assumption e(P ) = 1 and P ∈ Q[X, Y ] irreducible im-ply that P is irreducible in Q[X, Y ]. Indeed otherwise P would be the product of 2polynomials both vanishing at (0, 0), therefore ∂P

∂X and ∂P∂Y would both vanish there too.

Say for brevity p = degX P and q = degY P . Clearly we may assume p > 0. Let lbe as in the hypothesis, we note log l ≥ 1. We first show that P (l, Y ) /∈ Q. Indeed thisfollows easily for example by Lemma A.1(iii). We fix y ∈ Q with P (l, y) = 0. We mustshow D = [Q(y) : Q] = q. The inequality D ≤ q holds trivially so it suffices to showD ≥ q.

By Corollary B.1 and the hypothesis (B.4.1) we conclude lgcd(l, y) > 0 and so

(B.4.4) Dlgcd(l, y) ≥ log l

by (B.4.2).By Theorem B.3 and e(P ) = 1 we obtain

(B.4.5) lgcd(l, y)− 1qh(l) ≤ 183d max{d, hp(P )}1/2 max{1, h(l), h(y)}1/2.

Furthermore, h(l) = log l. In (B.4.5) we bound lgcd(l, y) from below with (B.4.4) andthen multiply with Dq ≤ q2 to get

(B.4.6) (q −D) log l ≤ 183dq2 max{d, hp(P )}1/2 max{1, log l, h(y)}1/2.

The proof follows by splitting up into two cases:First say log l ≥ h(y). Then (B.4.6) implies

(q −D)2 log l ≤ 1832d2q4 max{d, hp(P )}.This inequality contradicts (B.4.1) if q −D ≥ 1. Hence q ≤ D and we are done in thiscase.

Second say log l ≤ h(y). By Corollary A.1 and (B.4.1) we have h(y) ≤ 2pq log l.

Therefore max{1, log l, h(y)} ≤ 2pq log l. We insert this inequality in (B.4.6) to conclude

(B.4.7) (q −D)2 log l ≤ 7.104d2pq3 max{d, hp(P )}.As before, if q−D ≥ 1, then (B.4.7) is incompatible with (B.4.1). We conclude q ≤ D.�

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Curriculum Vitae

I was born on July 23rd 1978 in Schlieren, Zurich. I am citizen of Trub, Bern. Myparents are Erich and Annemarie Habegger.

Education

Oct.2003 - Jun.2007 PhD in Mathematics at the University ofBasel, Switzerland.Adviser: Prof. D. W. Masser.

Sep.2005 - Feb.2006 Research visit at the Insitut deMathematiques de Jussieu, Paris, France.

Oct.2000 - Apr.2003 Advanced studies in Mathematics at theUniversity of Basel.Degree: Diplom.

Oct.1998 - Oct.2000 Basic studies in Mathematics at theUniversity of Basel.Degree: Vordiplom.

Aug.1994 - Dec.1997 Gymnasium Oberwil, Baselland.Degree: Matura.

I have visited lectures and seminars of Prof. C. Bandle, Prof. Y. Bilu, Prof. H. Kraft,Dr. J. Lieberum, Prof. D. W.Masser, Dr. S. Mohrdieck, Prof. W. Reichel, Prof.B. Scarpellini, Prof. G. Tammann. Prof. F. Thielemann, Prof. H. Thomas, and Prof.D. Trautmann.

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Heights and multiplicative relations on algebraic varieties · varieties are defined over Q, the field of algebraic numbers. One can define a height function on the set of algebraic